14 PERIODIC MOTION 14.1 IDENTIFY: We want to relate the characteristics of various waves, such as the period, frequency and angular frequency SET UP: The frequency f in Hz is the number of cycles per second The angular frequency ω is ω = 2π f and has units of radians per second The period T is the time for one cycle of the wave and has units of seconds The period and frequency are related by T = EXECUTE: (a) T = f 1 = = 2.15 × 10−3 s f 466 Hz ω = 2π f = 2π (466 Hz) = 2.93 × 103 rad/s (b) f = 1 = = 2.00 × 104 Hz ω = 2π f = 1.26 × 105 rad/s T 50.0 × 10−6 s (c) f = 2.7 × 1015 rad/s ω = 4.3 × 1014 Hz to so f ranges from 2π rad 2π 4.7 × 1015 rad/s so T ranges from = 7.5 × 1014 Hz T = 2π rad f 14 7.5 × 10 Hz = 1.3 × 10−15 s to 4.3 × 1014 Hz = 2.3 × 10−15 s 1 = = 2.0 × 10−7 s and ω = 2π f = 2π (5.0 × 106 Hz) = 3.1 × 107 rad/s f 5.0 × 106 Hz EVALUATE: Visible light has much higher frequency than either sounds we can hear or ultrasound Ultrasound is sound with frequencies higher than what the ear can hear Large f corresponds to small T (d) T = 14.2 14.3 IDENTIFY and SET UP: The amplitude is the maximum displacement from equilibrium In one period the object goes from x = + A to x = − A and returns EXECUTE: (a) A = 0.120 m (b) 0.800 s = T/ so the period is 1.60 s (c) f = = 0.625 Hz T EVALUATE: Whenever the object is released from rest, its initial displacement equals the amplitude of its SHM 2π IDENTIFY: The period is the time for one vibration and ω = T SET UP: The units of angular frequency are rad/s EXECUTE: The period is 0.50 s = 1.14 × 10−3 s and the angular frequency is ω = 2π = 5.53 × 103 rad/s 440 T EVALUATE: There are 880 vibrations in 1.0 s, so f = 880 Hz This is equal to 1/T © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-1 14-2 14.4 14.5 14.6 Chapter 14 IDENTIFY: The period is the time for one cycle and the amplitude is the maximum displacement from equilibrium Both these values can be read from the graph SET UP: The maximum x is 10.0 cm The time for one cycle is 16.0 s EXECUTE: (a) T = 16.0 s so f = = 0.0625 Hz T (b) A = 10.0 cm (c) T = 16.0 s (d) ω = 2π f = 0.393 rad/s EVALUATE: After one cycle the motion repeats IDENTIFY: This displacement is 14 of a period SET UP: T = 1/f = 0.200 s EXECUTE: t = 0.0500 s EVALUATE: The time is the same for x = A to x = 0, for x = to x = − A, for x = − A to x = and for x = to x = A IDENTIFY: Apply Eq (14.12) SET UP: The period will be twice the interval between the times at which the glider is at the equilibrium position 2 ⎛ 2π ⎞ ⎛ 2π ⎞ EXECUTE: k = ω 2m = ⎜ ⎟ (0.200 kg) = 0.292 N/m ⎟ m=⎜ 2(2 60 s) T ⎝ ⎠ ⎝ ⎠ 14.7 EVALUATE: N = kg ⋅ m/s , so N/m = kg/s IDENTIFY and SET UP: Use Eq (14.1) to calculate T, Eq (14.2) to calculate ω and Eq (14.10) for m EXECUTE: (a) T = 1/f = 1/6.00 Hz = 0.167 s (b) ω = 2π f = 2π (6.00 Hz) = 37.7 rad/s (c) ω = k/m implies m = k/ω = (120 N/m)/(37.7 rad/s) = 0.0844 kg EVALUATE: We can verify that k/ω has units of mass 14.8 IDENTIFY: The mass and frequency are related by f = 2π k m k = constant, so f1 m1 = f m2 2π EXECUTE: (a) m1 = 0.750 kg, f1 = 1.33 Hz and m2 = 0.750 kg + 0.220 kg = 0.970 kg f m= SET UP: f = f1 0.750 kg m1 = (1.33 Hz) = 1.17 Hz 0.970 kg m2 (b) m2 = 0.750 kg − 0.220 kg = 0.530 kg f = (1.33 Hz) 14.9 0.750 kg = 1.58 Hz 0.530 kg EVALUATE: When the mass increases the frequency decreases and when the mass decreases the frequency increases IDENTIFY: For SHM the motion is sinusoidal SET UP: x(t ) = A cos(ωt ) 2π 2π = = 6.981 rad/s T 0.900 s (a) x = 0.320 m at t1 = Let t2 be the instant when x = 0.160 m Then we have EXECUTE: x(t ) = A cos(ωt ), where A = 0.320 m and ω = 0.160 m = (0.320 m) cos(ωt2 ) cos(ωt2 ) = 0.500 ωt2 = 1.047 rad t2 = 1.047 rad = 0.150 s It takes 6.981 rad/s t2 − t1 = 0.150 s © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion (b) Let t3 be when x = Then we have cos(ωt3 ) = and ωt3 = 1.571 rad t3 = 14.10 1.571 rad = 0.225 s It 6.981 rad/s takes t3 − t2 = 0.225 s − 0.150 s = 0.0750 s EVALUATE: Note that it takes twice as long to go from x = 0.320 m to x = 0.160 m than to go from x = 0.160 m to x = 0, even though the two distances are the same, because the speeds are different over the two distances IDENTIFY: For SHM the restoring force is directly proportional to the displacement and the system obeys Newton’s second law k SET UP: Fx = max and f = 2π m EXECUTE: Fx = ma x gives a x = − kx k a −5.30 m/s =− x =− = 18.93 s −2 , so m 0.280 m m x k 18.93 s −2 = 0.692 Hz = m 2π EVALUATE: The period is around 1.5 s, so this is a rather slow vibration IDENTIFY: Use Eq (14.19) to calculate A The initial position and velocity of the block determine φ x (t ) is given by Eq (14.13) SET UP: cosθ is zero when θ = ± π /2 and sin(π /2) = f = 14.11 14-3 2π EXECUTE: (a) From Eq (14.19), A = v0 ω v0 k/m = 0.98 m (b) Since x(0) = 0, Eq (14.14) requires φ = ± π2 Since the block is initially moving to the left, v0 x < and Eq (14.7) requires that sin φ > , so φ = + π2 (c) cos (ωt + (π /2)) = −sin ωt, so x = (−0.98 m) sin((12.2 rad/s)t ) EVALUATE: The x (t ) result in part (c) does give x = at t = and x < for t slightly greater than zero 14.12 IDENTIFY and SET UP: We are given k, m, x0 , and v0 Use Eqs (14.19), (14.18) and (14.13) EXECUTE: (a) Eq (14.19): A = x02 + v02x /ω = x02 + mv02x /k A = (0.200 m) + (2.00 kg)(− 4.00 m/s) /(300 N/m) = 0.383 m (b) Eq (14.18): φ = arctan( − v0 x /ω x0 ) ω = k/m = (300 N/m)/2.00 kg = 12.25 rad/s ⎛ ⎞ (−4.00 m/s) ⎟ = arctan(+1.633) = 58.5° (or 1.02 rad) (12.25 rad/s)(0.200 m) ⎝ ⎠ (c) x = A cos(ωt + φ ) gives x = (0.383 m)cos([12.2rad/s]t + 1.02 rad) EVALUATE: At t = the block is displaced 0.200 m from equilibrium but is moving, so A > 0.200 m According to Eq (14.15), a phase angle φ in the range < φ < 90° gives v0 x < φ = arctan ⎜ − 14.13 IDENTIFY: For SHM, a x = − ω x = − (2π f ) x Apply Eqs (14.13), (14.15) and (14.16), with A and φ from Eqs (14.18) and (14.19) SET UP: x = 1.1 cm, v0 x = −15 cm/s ω = 2π f , with f = 2.5 Hz EXECUTE: (a) a x = −(2π (2.5 Hz)) (1.1 × 10−2 m) = −2.71 m/s (b) From Eq (14.19) the amplitude is 1.46 cm, and from Eq (14.18) the phase angle is 0.715 rad The angular frequency is 2π f = 15.7 rad/s, so x = (1.46 cm) cos ((15.7 rad/s)t + 0.715 rad), vx = (−22.9 cm/s) sin ((15.7 rad/s)t + 0.715 rad) and a x = (−359 cm/s ) cos ((15.7 rad/s)t + 0.715 rad) EVALUATE: We can verify that our equations for x, vx and a x give the specified values at t = © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-4 Chapter 14 14.14 IDENTIFY: The motion is SHM, and in each case the motion described is one-half of a complete cycle 2π SET UP: For SHM, x = A cos(ωt ) and ω = T EXECUTE: (a) The time is half a period The period is independent of the amplitude, so it still takes 2.70 s 2π (b) x = 0.090 m at time t1 T = 5.40 s and ω = = 1.164 rad/s x1 = A cos(ωt1 ) cos(ωt1 ) = 0.500 T ωt1 = 1.047 rad and t1 = 0.8997 s x = −0.090 m at time t2 cos(ωt2 ) = −0.500 m ωt2 = 2.094 rad and t2 = 1.800 s The elapsed time is t2 − t1 = 1.800 s − 0.8997 s = 0.900 s EVALUATE: It takes less time to travel from ±0.090 m in (b) than it originally did because the block has larger speed at ± 0.090 m with the increased amplitude 14.15 m Use the information about the empty chair to calculate k k SET UP: When m = 42.5 kg, T = 1.30 s IDENTIFY: Apply T = 2π EXECUTE: Empty chair: T = 2π m 4π 2m 4π (42.5 kg) = = 993 N/m gives k = k T2 (1.30 s) m T 2k (2.54 s) (993 N/m) = 162 kg and gives m = = k 4π 4π = 162 kg − 42.5 kg = 120 kg With person in chair: T = 2π mperson 14.16 EVALUATE: For the same spring, when the mass increases, the period increases IDENTIFY and SET UP: Use Eq (14.12) for T and Eq (14.4) to relate a x and k EXECUTE: T = 2π m , m = 0.400 kg k Use a x = − 2.70 m/s to calculate k: − kx = max gives k =− m ma x (0.400 kg)(− 2.70 m/s ) =− = + 3.60 N/m T = 2π = 2.09 s k x 0.300 m EVALUATE: a x is negative when x is positive ma x /x has units of N/m and 14.17 m/k has units of s m k k a x = − x so amax = A F = − kx k m m SET UP: a x is proportional to x so a x goes through one cycle when the displacement goes through one IDENTIFY: T = 2π cycle From the graph, one cycle of a x extends from t = 0.10 s to t = 0.30 s, so the period is T = 0.20 s k = 2.50 N/cm = 250 N/m From the graph the maximum acceleration is 12.0 m/s EXECUTE: (a) T = 2π 2 m ⎛ T ⎞ ⎛ 0.20 s ⎞ gives m = k ⎜ ⎟ = (250 N/m) ⎜ ⎟ = 0.253 kg k π ⎝ ⎠ ⎝ 2π ⎠ mamax (0.253 kg)(12.0 m/s ) = = 0.0121 m = 1.21 cm k 250 N/m (c) Fmax = kA = (250 N/m)(0.0121 m) = 3.03 N (b) A = EVALUATE: We can also calculate the maximum force from the maximum acceleration: Fmax = mamax = (0.253 kg)(12.0 m/s ) = 3.04 N, which agrees with our previous results 14.18 IDENTIFY: The general expression for vx (t ) is vx (t ) = −ω A sin(ωt + φ ) We can determine ω and A by comparing the equation in the problem to the general form SET UP: ω = 4.71 rad/s ω A = 3.60 cm/s = 0.0360 m/s 2π 2π rad EXECUTE: (a) T = = = 1.33 s ω 4.71 rad/s © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion (b) A = 0.0360 m/s ω = 14-5 0.0360 m/s = 7.64 × 10−3 m = 7.64 mm 4.71 rad/s (c) amax = ω A = (4.71 rad/s) (7.64 × 10−3 m) = 0.169 m/s k so k = mω = (0.500 kg)(4.71 rad/s) = 11.1 N/m m EVALUATE: The overall positive sign in the expression for vx (t ) and the factor of −π /2 both are related to the phase factor φ in the general expression IDENTIFY: Compare the specific x (t ) given in the problem to the general form of Eq (14.13) SET UP: A = 7.40 cm, ω = 4.16 rad/s, and φ = −2.42 rad (d) ω = 14.19 EXECUTE: (a) T = (b) ω = (c) vmax 2π ω = 2π = 1.51 s 4.16 rad/s k so k = mω = (1.50 kg)(4.16 rad/s) = 26.0 N/m m = ω A = (4.16 rad/s)(7.40 cm) = 30.8 cm/s (d) Fx = − kx so Fmax = kA = (26.0 N/m)(0.0740 m) = 1.92 N (e) x (t ) evaluated at t = 1.00 s gives x = −0.0125 m vx = −ω A sin(ωt + φ ) = 30.4 cm/s a x = − kx/m = −ω x = +0.216 m/s (f) Fx = − kx = − (26.0 N/m)( − 0.0125 m) = + 0.325 N 14.20 EVALUATE: The maximum speed occurs when x = and the maximum force is when x = ± A IDENTIFY: The frequency of vibration of a spring depends on the mass attached to the spring Differences in frequency are due to differences in mass, so by measuring the frequencies we can determine the mass of the virus, which is the target variable k SET UP: The frequency of vibration is f = 2π m Solve: (a) The frequency without the virus is fs = fs + v = ⎛ k f s+ v = ⎜ ⎜ fs ms + mv ⎝ 2π 2π k ms + mv 2π k , and the frequency with the virus is ms ⎞⎛ ms ⎞ 2π ⎟⎜ ⎟= ⎟⎜ k ⎠⎟ ⎠⎝ ms = ms + mv + mv /ms ⎛ f ⎞ Solving for mv gives (b) ⎜ s + v ⎟ = ⎝ fs ⎠ + mv /ms ⎛⎡ ⎞ ⎛ ⎡ f ⎤2 ⎞ 2.00 × 1015 Hz ⎤ ⎟ = 9.99 × 10−15 g, or mv = ms ⎜ ⎢ s ⎥ − 1⎟ = (2.10 × 10−16 g) ⎜ ⎢ − ⎥ ⎜ ⎢ 2.87 × 1014 Hz ⎥ ⎟ ⎜ ⎣ fs + v ⎦ ⎟ ⎦ ⎝ ⎠ ⎝⎣ ⎠ mv = 9.99 femtograms 14.21 EVALUATE: When the mass increases, the frequency of oscillation increases IDENTIFY and SET UP: Use Eqs (14.13), (14.15) and (14.16) EXECUTE: f = 440 Hz, A = 3.0 mm, φ = (a) x = A cos(ωt + φ ) ω = 2π f = 2π (440 Hz) = 2.76 × 103 rad/s x = (3.0 × 10−3 m)cos((2.76 × 103 rad/s)t ) (b) vx = −ω A sin(ωt + φ ) vmax = ω A = (2.76 × 103 rad/s)(3.0 × 10−3 m) = 8.3 m/s (maximum magnitude of velocity) a x = −ω A cos(ωt + φ ) © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-6 Chapter 14 amax = ω A = (2.76 × 103 rad/s) (3.0 × 10−3 m) = 2.3 × 104 m/s (maximum magnitude of acceleration) (c) a x = −ω A cos ωt da x /dt = +ω A sin ωt = [2π (440 Hz)]3 (3.0 × 10−3 m)sin([2.76 × 103 rad/s]t ) = (6.3 × 107 m/s3 )sin([2.76 × 103 rad/s]t ) Maximum magnitude of the jerk is ω A = 6.3 × 107 m/s3 14.22 EVALUATE: The period of the motion is small, so the maximum acceleration and jerk are large IDENTIFY: The mechanical energy of the system is conserved The maximum acceleration occurs at the maximum displacement and the motion is SHM m kA SET UP: Energy conservation gives mvmax , and amax = = kA2 , T = 2π k m 2 = kA2 gives EXECUTE: (a) From the graph, we read off T = 16.0 s and A = 10.0 cm = 0.100 m mvmax 2 k m T = 2π , so m k vmax = A k 2π ⎛ 2π = Therefore vmax = A ⎜ m T ⎝ T ⎞ ⎛ 2π ⎞ ⎟ = (0.100 m) ⎜ ⎟ = 0.0393 m/s ⎠ ⎝ 16.0 s ⎠ kA ⎛ 2π ⎞ ⎛ 2π ⎞ =⎜ ⎟ A=⎜ ⎟ (0.100 m) = 0.0154 m/s m ⎝ T ⎠ ⎝ 16.0 s ⎠ EVALUATE: The acceleration is much less than g IDENTIFY: The mechanical energy of the system is conserved The maximum acceleration occurs at the maximum displacement and the motion is SHM kA SET UP: Energy conservation gives mvmax = kA2 and amax = m 2 (b) amax = 14.23 EXECUTE: A = 0.120 m 2 k ⎛ vmax ⎞ ⎛ 3.90 m/s ⎞ −2 mvmax = kA2 gives =⎜ ⎟ =⎜ ⎟ = 1056 s 2 m ⎝ A ⎠ ⎝ 0.120 m ⎠ kA = (1056 s −2 )(0.120 m) = 127 m/s m EVALUATE: The acceleration is much greater than g IDENTIFY: The mechanical energy of the system is conserved, Newton’s second law applies and the motion is SHM 1 SET UP: Energy conservation gives mvx2 + kx = kA2 , Fx = ma x , Fx = − kx, and the period is 2 amax = 14.24 T = 2π m k EXECUTE: Solving m k 2 mvx + kx = kA for vx gives vx = , so A2 − x T = 2π k m 2 k 2π 2π = = = 1.963 s −1 vx = (1.963 s −1) (0.250 m) − (0.160 m) = 0.377 m/s m T 3.20 s kx a x = − = −(1.963 s −1 ) (0.160 m) = −0.617 m/s m EVALUATE: The block is on the positive side of the equilibrium position ( x = 0) and is moving in the 14.25 positive direction but is accelerating in the negative direction, so it must be slowing down IDENTIFY: vmax = ω A = 2π fA K max = 12 mvmax SET UP: The fly has the same speed as the tip of the tuning fork EXECUTE: (a) vmax = 2π fA = 2π (392 Hz)(0.600 × 10−3 m) = 1.48 m/s (b) K max = 12 mvmax = 12 (0.0270 × 10−3 kg)(1.48 m/s)2 = 2.96 × 10−5 J EVALUATE: vmax is directly proportional to the frequency and to the amplitude of the motion © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14.26 14-7 IDENTIFY and SET UP: Use Eq (14.21) to relate K and U U depends on x and K depends on vx EXECUTE: (a) U + K = E , so U = K says that 2U = E ( 12 kx2 ) = 12 kA2 and x = ± A/ 2; magnitude is A/ But U = K also implies that 2K = E ( 12 mv2 ) = 12 kA2 and v x x = ± k/m A/ = ±ω A/ 2; magnitude is ω A/ (b) In one cycle x goes from A to to − A to to + A Thus x = + A twice and x = − A/ twice in each cycle Therefore, U = K four times each cycle The time between U = K occurrences is the time Δta for x1 = + A/ to x2 = − A 2, time Δtb for x1 = − A/ to x2 = + A/ 2, time Δtc for x1 = + A/ to x2 = + A 2, or the time Δtd for x1 = − A/ to x2 = − A/ 2, as shown in Figure 14.26 Δta = Δtb Δtc = Δtd Figure 14.26 Calculation of Δta : Specify x in x = A cos ωt (choose φ = so x = A at t = ) and solve for t x1 = + A/ implies A/ = A cos(ωt1 ) cos ωt1 = 1/ so ωt1 = arccos(1/ 2) = π /4 rad t1 = π /4ω x2 = − A/ implies − A/ = A cos(ωt2 ) cos ωt2 = − 1/ so ωt1 = 3π /4 rad t2 = 3π /4ω Δta = t2 − t1 = 3π /4ω − π /4ω = π /2ω (Note that this is T/4, one fourth period.) Calculation of Δtd : x1 = − A/ implies t1 = 3π /4ω x2 = − A/ , t2 is the next time after t1 that gives cos ωt2 = − 1/ Thus ωt2 = ωt1 + π /2 = 5π /4 and t2 = 5π /4ω Δtd = t2 − t1 = 5π /4ω − 3π /4ω = π /2ω , so is the same as Δta Therfore the occurrences of K = U are equally spaced in time, with a time interval between them of π /2ω EVALUATE: This is one-fourth T, as it must be if there are equally spaced occurrences each period (c) EXECUTE: x = A/2 and U + K = E K = E − U = 12 kA2 − 12 kx = 12 kA2 − 12 k ( A/2) = 12 kA2 − 81 kA2 = 3kA2 /8 Then K 3kA2 /8 U kA = = and = = kA2 E kA2 E 2 EVALUATE: At x = all the energy is kinetic and at x = ± A all the energy is potiential But K = U does not occur at x = ± A/2, since U is not linear in x © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-8 Chapter 14 14.27 IDENTIFY: Velocity and position are related by E = 12 kA2 = 12 mvx2 + 12 kx Acceleration and position are related by − kx = ma x SET UP: The maximum speed is at x = and the maximum magnitude of acceleration is at x = ± A EXECUTE: (a) For x = 0, (b) vx = ± mv max = 12 kA2 and vmax = A k 450 N/m = (0.040 m) = 1.20 m/s m 0.500 kg k 450 N/m A2 − x = ± (0.040 m) − (0.015 m) = ± 1.11 m/s m 0.500 kg The speed is v = 1.11 m/s (c) For x = ± A, amax = (d) a x = − ⎛ 450 N/m ⎞ k A=⎜ ⎟ (0.040 m) = 36 m/s m 500 kg ⎝ ⎠ kx (450 N/m)(− 0.015 m) =− = +13.5 m/s m 0.500 kg (e) E = 12 kA2 = 12 (450 N/m)(0.040 m) = 0.360 J EVALUATE: The speed and acceleration at x = − 0.015 m are less than their maximum values 14.28 IDENTIFY and SET UP: a x is related to x by Eq (14.4) and vx is related to x by Eq (14.21) a x is a maximum when x = ± A and vx is a maximum when x = t is related to x by Eq (14.13) EXECUTE: (a) − kx = ma x so a x = −( k/m) x (Eq.14.4) But the maximum x is A, so amax = (k /m) A = ω A f = 0.850 Hz implies ω = k /m = 2π f = 2π (0.850 Hz) = 5.34 rad/s amax = ω A = (5.34 rad/s) (0.180 m) = 5.13 m/s mv x + 12 kx = 12 kA2 vx = vmax when x = so mv max = 12 kA2 vmax = k/m A = ω A = (5.34 rad/s)(0.180 m) = 0.961 m/s (b) a x = −( k/m) x = −ω x = − (5.34 rad/s)2 (0.090 m) = −2.57 m/s mv x + 12 kx = 12 kA2 says that vx = ± k /m A2 − x = ± ω A2 − x vx = ± (5.34 rad/s) (0.180 m)2 − (0.090 m)2 = ± 0.832 m/s The speed is 0.832 m/s (c) x = A cos(ωt + φ ) Let φ = − π /2 so that x = at t = Then x = A cos(ωt − π /2) = A sin(ωt ) [Using the trig identity cos( a − π /2) = sin a ] Find the time t that gives x = 0.120 m 0.120 m = (0.180 m)sin(ωt ) sin ωt = 0.6667 t = arcsin(0.6667)/ω = 0.7297 rad/(5.34 rad/s)=0.137 s EVALUATE: It takes one-fourth of a period for the object to go from x = to x = A = 0.180 m So the time we have calculated should be less than T /4 T = 1/f = 1/0.850 Hz = 1.18 s, T /4 = 0.295 s, and the time we calculated is less than this Note that the a x and vx we calculated in part (b) are smaller in magnitude than the maximum values we calculated in part (b) (d) The conservation of energy equation relates v and x and F = ma relates a and x So the speed and acceleration can be found by energy methods but the time cannot Specifying x uniquely determines a x but determines only the magnitude of vx ; at a given x the object could be moving either in the + x or − x direction © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14.29 14-9 IDENTIFY: Use the results of Example 14.5 and also that E = 12 kA2 SET UP: In the example, A2 = A1 M and now we want A2 = 12 A1 Therefore, = M +m m = 3M For the energy, E2 = 12 kA22 , but since A2 = 12 A1, E2 = 14 E1, and E M , or M +m is lost to heat EXECUTE: The putty and the moving block undergo a totally inelastic collision and the mechanical energy of the system decreases 14.30 IDENTIFY and SET UP: Use Eq (14.21) x = ± Aω when vx = and vx = ± vmax when x = EXECUTE: (a) E = 12 mv + 12 kx E = 12 (0.150 kg)(0.300 m/s)2 + 12 (300 N/m)(0.012 m)2 = 0.0284 J (b) E = 12 kA2 so A = E /k = 2(0.0284 J)/300 N/m = 0.014 m (c) E = 12 mvmax so vmax = E /m = 2(0.0284 J)/0.150 kg = 0.615 m/s 14.31 EVALUATE: The total energy E is constant but is transferred between kinetic and potential energy during the motion IDENTIFY: Conservation of energy says 12 mv + 12 kx = 12 kA2 and Newton’s second law says − kx = ma x SET UP: Let + x be to the right Let the mass of the object be m EXECUTE: k = − ⎛ −8.40 m/s ⎞ ma x −2 = −m ⎜ ⎟ = (14.0 s ) m 600 m x ⎝ ⎠ ⎛ ⎞ m A = x (m/k )v = (0.600 m) + ⎜⎜ ⎟⎟ (2.20 m/s) = 0.840 m The object will therefore −2 [14 s ] m ⎝ ⎠ travel 0.840 m − 0.600 m = 0.240 m to the right before stopping at its maximum amplitude 14.32 EVALUATE: The acceleration is not constant and we cannot use the constant acceleration kinematic equations IDENTIFY: When the box has its maximum speed all of the energy of the system is in the form of kinetic energy When the stone is removed the oscillating mass is decreased and the speed of the remaining mass m is unchanged The period is given by T = 2π k SET UP: The maximum speed is vmax = ω A = k A With the stone in the box m = 8.64 kg and m A = 0.0750 m EXECUTE: (a) T = 2π m 5.20 kg = 2π = 0.740 s k 375 N/m (b) Just before the stone is removed, the speed is vmax = 375 N/m (0.0750 m) = 0.494 m/s The speed of 8.64 kg the box isn’t altered by removing the stone but the mass on the spring decreases to 5.20 kg The new m 5.20 kg vmax = (0.494 m/s) = 0.0582 m The new amplitude can also be calculated amplitude is A = k 375 N/m as 5.20 kg (0.0750 m) = 0.0582 m 8.64 kg (c) T = 2π m The force constant remains the same m decreases, so T decreases k © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-10 Chapter 14 EVALUATE: After the stone is removed, the energy left in the system is 1m v = 12 (5.20 kg)(0.494 m/s) = 0.6345 J This then is the energy stored in the spring at its box max maximum extension or compression and 14.33 kA2 = 0.6345 J This gives the new amplitude to be 0.0582 m, in agreement with our previous calculation IDENTIFY: The mechanical energy (the sum of the kinetic energy and potential energy) is conserved SET UP: K + U = E , with E = 12 kA2 and U = 12 kx EXECUTE: U = K says 2U = E This gives 2( 12 kx ) = 12 kA2 , so x = A/ EVALUATE: When x = A/2 the kinetic energy is three times the elastic potential energy 14.34 IDENTIFY: The velocity is a sinusoidal function From the graph we can read off the period and use it to calculate the other quantities SET UP: The period is the time for cycle; after time T the motion repeats The graph shows that T = 1.60 s and vmax = 20.0 cm/s Mechanical energy is conserved, so 12 mvx2 + 12 kx = 12 kA2 , and Newton’s second law applies to the mass EXECUTE: (a) T = 1.60 s (from the graph) = 0.625 Hz T (c) ω = 2π f = 3.93 rad/s (b) f = (d) vx = vmax when x = so kA2 2 = 12 mvmax A = vmax graph in the problem, vmax = 0.20 m/s, so A = m f = k 2π k so A = vmax /(2π f ) From the m 0.20 m/s = 0.051 m = 5.1 cm The mass is at x = ± A 2π (0.625 Hz) when vx = 0, and this occurs at t = 0.4 s, 1.2 s, and 1.8 s (e) Newton’s second law gives − kx = ma x , so kA = (2π f ) A = (4π )(0.625 Hz)2 (0.051 m) = 0.79 m/s = 79 cm/s The acceleration is m maximum when x = ± A and this occurs at the times given in (d) amax = 2 m ⎛ T ⎞ ⎛ 1.60 s ⎞ so m = k ⎜ ⎟ = (75 N/m) ⎜ ⎟ = 4.9 kg k π ⎝ ⎠ ⎝ 2π ⎠ EVALUATE: The speed is maximum at x = 0, when a x = The magnitude of the acceleration is (f) T = 2π maximum at x = ± A, where vx = 14.35 IDENTIFY: Work in an inertial frame moving with the vehicle after the engines have shut off The acceleration before engine shut-off determines the amount the spring is initially stretched The initial speed of the ball relative to the vehicle is zero SET UP: Before the engine shut-off the ball has acceleration a = 5.00 m/s EXECUTE: (a) Fx = − kx = ma x gives A = ma (3.50 kg)(5.00 m/s ) = = 0.0778 m This is the amplitude k 225 N/m of the subsequent motion k 225 N/m (b) f = = = 1.28 Hz 2π m 2π 3.50 kg (c) Energy conservation gives kA2 2 = 12 mvmax and vmax = k 225 N/m A= (0.0778 m) = 0.624 m/s m 3.50 kg EVALUATE: During the simple harmonic motion of the ball its maximum acceleration, when x = ± A, continues to have magnitude 5.00 m/s © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14.68 14-19 EVALUATE: For a given amplitude, the maximum acceleration and maximum velocity increase when the frequency of the motion increases and the period decreases m IDENTIFY: T = 2π The period changes when the mass changes k SET UP: M is the mass of the empty car and the mass of the loaded car is M = 250 kg EXECUTE: The period of the empty car is TE = 2π TL = 2π M The period of the loaded car is k M + 250 kg (250 kg)(9.80 m/s ) k= = 6.125 × 104 N/m k 4.00 × 10−2 m 2 ⎛T ⎞ ⎛ 1.92 s ⎞ M = ⎜ L ⎟ k − 250 kg = ⎜ ⎟ (6.125 × 10 N/m) − 250 kg = 5.469 × 10 kg 2 π π ⎝ ⎠ ⎝ ⎠ TE = 2π 14.69 5.469 × 103 kg 6.125 × 104 N/m = 1.88 s EVALUATE: When the mass decreases, the period decreases IDENTIFY and SET UP: Use Eqs (14.12), (14.21) and (14.22) to relate the various quantities to the amplitude EXECUTE: (a) T = 2π m/k ; independent of A so period doesn’t change f = 1/T ; doesn’t change ω = 2π f ; doesn’t change (b) E = 12 kA2 when x = ± A When A is halved E decreases by a factor of 4; E2 = E1 /4 (c) vmax = ω A = 2π fA vmax ,1 = 2π fA1, vmax ,2 = 2π fA2 (f doesn’t change) Since A2 = 12 A1 , vmax ,2 = 2π f ( 12 A1 ) = 12 2π fA1 = 12 vmax,1; vmax is one-half as great (d) vx = ± k/m A2 − x x = ± A1 /4 gives vx = ± k /m A2 − A12 /16 With the original amplitude v1x = ± k/m A12 − A12 /16 = ± 15/16( k/m ) A1 With the reduced amplitude v2 x = ± k/m A22 − A12 /16 = ± k/m ( A1/2) − A12 /16 = ± 3/16( k/m ) A1 v1x /v2 x = 15/3 = 5, so v2 = v1 / 5; the speed at this x is 1/ times as great (e) U = 12 kx ; same x so same U K = 12 mvx2 ; K1 = 12 mv12x K = 12 mv22x = 12 m(v1x / 5) = 15 ( 12 mv12x ) = K1 /5; 1/5 times as great 14.70 EVALUATE: Reducing A reduces the total energy but doesn’t affect the period and the frequency (a) IDENTIFY and SET UP: Combine Eqs (14.12) and (14.21) to relate vx and x to T EXECUTE: T = 2π m /k We are given information about vx at a particular x The expression relating these two quantities comes from conservation of energy: mv x We can solve this equation for m = k + 12 kx = 12 kA2 m /k , and then use that result to calculate T mvx2 = k ( A2 − x ) gives A2 − x (0.100 m)2 − (0.060 m)2 = = 0.200 s vx 0.400 m/s © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-20 Chapter 14 Then T = 2π m /k = 2π (0.200 s) = 1.26 s (b) IDENTIFY and SET UP: We are asked to relate x and vx , so use conservation of energy equation: mv x 2 + 12 kx = 12 kA2 kx = kA2 − mvx2 x = A2 − (m /k )vx2 = (0.100 m)2 − (0.200 s) (0.160 m/s) = 0.0947 m EVALUATE: Smaller vx means larger x (c) IDENTIFY: If the slice doesn’t slip, the maximum acceleration of the plate (Eq.14.4) equals the maximum acceleration of the slice, which is determined by applying Newton’s second law to the slice SET UP: For the plate, − kx = max and ax = −(k/m) x The maximum | x | is A, so amax = (k /m) A If the carrot slice doesn’t slip then the static friction force must be able to give it this much acceleration The free-body diagram for the carrot slice (mass m′ ) is given in Figure 14.70 EXECUTE: ∑ Fy = ma y n − m′g = n = m′g Figure 14.70 ∑ Fx = ma x μ s n = m′a μ s m′g = m′a and a = μ s g But we require that a = amax = (k /m) A = μ s g and μ s = k A ⎛ ⎞ ⎛ 0.100 m ⎞ =⎜ ⎟ ⎜ ⎟ = 0.255 m g ⎝ 0.200 s ⎠ ⎝ 9.80 m/s ⎠ EVALUATE: We can write this as μs = ω A/g More friction is required if the frequency or the amplitude 14.71 is increased IDENTIFY: The largest downward acceleration the ball can have is g whereas the downward acceleration of the tray depends on the spring force When the downward acceleration of the tray is greater than g, then the ball leaves the tray y (t ) = A cos(ωt + φ ) SET UP: The downward force exerted by the spring is F = kd , where d is the distance of the object above F kd the equilibrium point The downward acceleration of the tray has magnitude = , where m is the total m m mass of the ball and tray x = A at t = 0, so the phase angle φ is zero and + x is downward mg (1.775 kg)(9.80 m/s ) kd = = 9.40 cm This point is 9.40 cm above the = g gives d = k 185 N/m m equilibrium point so is 9.40 cm + 15.0 cm = 24.4 cm above point A EXECUTE: (a) (b) ω = k 185 N/m = = 10.2 rad/s The point in (a) is above the equilibrium point so x = − 9.40 cm m 1.775 kg ⎛ −9.40 cm ⎞ 2.25 rad ⎛x⎞ x = A cos(ωt ) gives ωt = arccos ⎜ ⎟ = arccos ⎜ = 0.221 s ⎟ = 2.25 rad t = 10.2 rad/s ⎝ A⎠ ⎝ 15.0 cm ⎠ (c) kx 2 + 12 mv = 12 kA2 gives v = k 185 N/m ( A − x2 ) = ([0.150 m]2 − [− 0.0940 m]2 ) = 1.19 m/s m 1.775 kg © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14-21 m = 0.615 s To go from the lowest point to the highest point takes k time T /2 = 0.308 s The time in (b) is less than this, as it should be G G k IDENTIFY: In SHM, amax = A Apply ∑ F = ma to the top block mtot EVALUATE: The period is T = 2π 14.72 SET UP: The maximum acceleration of the lower block can’t exceed the maximum acceleration that can be given to the other block by the friction force EXECUTE: For block m, the maximum friction force is fs = μs n = μs mg ∑ Fx = ma x gives μs mg = ma and a = μs g Then treat both blocks together and consider their simple harmonic motion 14.73 ⎛ k ⎞ ⎛ k ⎞ μs g ( M + m ) amax = ⎜ ⎟ A Set amax = a and solve for A: μs g = ⎜ ⎟ A and A = k ⎝M + m⎠ ⎝M +m⎠ EVALUATE: If A is larger than this the spring gives the block with mass M a larger acceleration than friction can give the other block, and the first block accelerates out from underneath the other block IDENTIFY: Apply conservation of linear momentum to the collision and conservation of energy to the k and T = motion after the collision f = 2π m f SET UP: The object returns to the equilibrium position in time T /2 EXECUTE: (a) Momentum conservation during the collision: mv0 = (2m)V 1 V = v0 = (2.00 m s) = 1.00 m s 2 Energy conservation after the collision: x= MV = k ω = 2π f = (20.0 kg)(1.00 m/s) = 0.426 m (amplitude) 110.0 N/m k /M f = 1 110.0 N/m 1 k/M = = 0.373 Hz T = = = 2.68 s 2π 2π 20.0 kg f 0.373 Hz (b) It takes 1/2 period to first return: 14.74 1 MV = kx 2 (2.68 s) = 1.34 s EVALUATE: The total mechanical energy of the system determines the amplitude The frequency and period depend only on the force constant of the spring and the mass that is attached to the spring IDENTIFY: The upward acceleration of the rocket produces an effective downward acceleration for objects in its frame of reference that is equal to g ′ = a + g SET UP: The amplitude is the maximum displacement from equilibrium and is unaffected by the motion L of the rocket The period is affected and is given by T = 2π g′ EXECUTE: The amplitude is 8.50° T = 2π 1.10 m 4.00 m/s + 9.80 m/s = 1.77 s EVALUATE: For a pendulum of the same length and with its point of support at rest relative to the earth, L T = 2π = 2.11 s The upward acceleration decreases the period of the pendulum If the rocket were g 14.75 instead accelerating downward, the period would be greater than 2.11 s IDENTIFY and SET UP: The bounce frequency is given by Eq (14.11) and the pendulum frequency by Eq (14.33) Use the relation between these two frequencies that is specified in the problem to calculate the equilibrium length L of the spring, when the apple hangs at rest on the end of the spring k EXECUTE: vertical SHM: f b = 2π m © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-22 Chapter 14 pendulum motion (small amplitude): f p = 2π g L The problem specifies that f p = 12 f b g 1 k = 2π L 2π m g /L = k /4m so L = gm /k = w/k = 4(1.00 N)/1.50 N/m = 2.67 m EVALUATE: This is the stretched length of the spring, its length when the apple is hanging from it (Note: Small angle of swing means v is small as the apple passes through the lowest point, so arad is small and the component of mg perpendicular to the spring is small Thus the amount the spring is stretched changes very little as the apple swings back and forth.) IDENTIFY: Use Newton’s second law to calculate the distance the spring is stretched from its unstretched length when the apple hangs from it SET UP: The free-body diagram for the apple hanging at rest on the end of the spring is given in Figure14.75 EXECUTE: ∑ Fy = ma y k ΔL − mg = ΔL = mg/k = w/k = 1.00 N/1.50 N/m = 0.667 m Figure 14.75 Thus the unstretched length of the spring is 2.67 m − 0.67 m = 2.00 m 14.76 EVALUATE: The spring shortens to its unstretched length when the apple is removed IDENTIFY: The vertical forces on the floating object must sum to zero The buoyant force B applied to the object by the liquid is given by Archimedes’s principle The motion is SHM if the net force on the object is of the form Fy = − ky and then T = 2π m/k SET UP: Take +y to be downward EXECUTE: (a) Vsubmerged = LA, where L is the vertical distance from the surface of the liquid to the bottom of the object Archimedes’ principle states ρ gLA = Mg , so L = M ρA (b) The buoyant force is ρ gA( L + y ) = Mg + F , where y is the additional distance the object moves downward Using the result of part (a) and solving for y gives y = F ρgA (c) The net force is Fnet = Mg − ρ gA( L + y ) = − ρ gAy k = ρ gA, and the period of oscillation is T = 2π 14.77 M M = 2π k ρ gA EVALUATE: The force F determines the amplitude of the motion but the period does not depend on how much force was applied IDENTIFY: Apply the results of Problem 14.76 SET UP: The additional force F applied to the buoy is the weight w = mg of the man w mg m (70.0 kg) EXECUTE: (a) y = = = = = 0.107 m ρ gA ρ gA ρ A (1.03 × 103 kg/m3 )π (0.450 m)2 (b) Note that in part (c) of Problem 14.76, M is the mass of the buoy, not the mass of the man, and A is the cross-section area of the buoy, not the amplitude The period is then © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion T = 2π 14.78 (950 kg) 3 (1.03 × 10 kg/m )(9.80 m/s )π (0.450 m)2 14-23 = 2.42 s EVALUATE: The period is independent of the mass of the man IDENTIFY: Tarzan on the swinging vine (with or without the chimp) is a simple pendulum SET UP: Tarzan first comes to rest after beginning his swing at the end of one-half of a cycle, so the period is T = 8.0 s Apply conservation of linear momentum to find the speed and kinetic energy of the system just after Tarzan has grabbed the chimp The figure in the solution to Problem 14.58 shows that the height h above the lowest point of the swing is h = L(1− cosθ ) The period of a simple pendulum is L g T = 2π EXECUTE: (a) T = 2π 2 L ⎛ T ⎞ ⎛ 8.0 s ⎞ so L = g ⎜ ⎟ = (9.80 m/s ) ⎜ ⎟ = 15.9 m g ⎝ 2π ⎠ ⎝ 2π ⎠ 1 = = 0.125 Hz The amplitude is 12° T 8.0 s (c) Apply conservation of energy to find Tarzan’s speed just before he grabs the chimp: U1 = K (b) f = mgL(1 − cosθ ) = 12 mv v = gL(1 − cosθ ) = 2(9.80 m/s )(15.9 m)(1 − cos12°) = 2.61 m/s Apply conservation of momentum to the inelastic collision between Tarzan and the chimp: (65 kg)(2.61 m/s) = (65 kg + 35 kg)V gives V = 1.70 m/s Apply conservation of energy to find the maximum angle of swing after the collision: 1m V2 tot = mtot gL(1 − cosθ ) Solving for θ gives g V (1.70 m/s) The length doesn’t change = = 0.00927 so θ = 7.8° f = 2π L gL 2(9.80 m/s )(15.9 m) so f remains 0.125 Hz f doesn’t depend on the mass or on the amplitude of swing − cosθ = 14.79 EVALUATE: Since the amplitude of swing is fairly small, we can use the small-angle approximation for which the period is independent of the amplitude If the angle of swing were a bit larger, this approximation would not be valid mobject gd Use the parallel-axis IDENTIFY: The object oscillates as a physical pendulum, so f = 2π I theorem, I = I cm + Md , to find the moment of inertia of each stick about an axis at the hook SET UP: The center of mass of the square object is at its geometrical center, so its distance from the hook is L cos 45° = L / The center of mass of each stick is at its geometrical center For each stick, mL2 I cm = 12 EXECUTE: The parallel-axis theorem gives I for each stick for an axis at the center of the square to be mL2 12 + m ( L /2 ) = 13 mL2 and the total I for this axis is mL2 For the entire object and an axis at the hook, applying the parallel-axis theorem again to the object of mass 4m gives I = 43 mL2 + 4m( L / 2) = 10 mL2 f = 14.80 2π mobject gd I = 2π 4mobject gL/ 10 m L2 object = ⎛ ⎜ ⎝ 2π ⎛ g⎞ = 0.921⎜ L ⎟⎠ ⎝ 2π g⎞ L ⎟⎠ EVALUATE: Just as for a simple pendulum, the frequency is independent of the mass A simple pendulum g of length L has frequency f = and this object has a frequency that is slightly less than this 2π L IDENTIFY: Conservation of energy says K + U = E SET UP: U = 12 kx and E = U max = 12 kA2 © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-24 Chapter 14 EXECUTE: (a) The graph is given in Figure 14.80 The following answers are found algebraically, to be used as a check on the graphical method 2E 2(0.200 J) = = 0.200 m (b) A = k (10.0 N/m) E = 0.050 J A (d) U = E x = = 0.141 m 2 (c) (e) From Eq (14.18), using v0 = (2 K /m) v K0 K0 2U and x0 = − , − = = = 0.429 m k ω x0 U0 (k /m) (2U /k ) and φ = arctan( 0.429) = 3.72 rad EVALUATE: The dependence of U on x is not linear and U = 12 U max does not occur at x = 12 xmax Figure 14.80 14.81 m so the period changes because the mass changes k dm dT = −2.00 × 10−3 kg/s The rate of change of the period is dt dt IDENTIFY: T = 2π SET UP: EXECUTE: (a) When the bucket is half full, m = 7.00 kg T = 2π (b) 7.00 kg = 1.49 s 125 N/m dT 2π d 1/ 2π −1/ dm π dm (m ) = m = = dt dt k dt k mk dt π dT dT is negative; the period is = ( −2.00 × 10−3 kg/s) = −2.12 × 10−4 s per s dt dt (7.00 kg)(125 N/m) getting shorter (c) The shortest period is when all the water has leaked out and m = 2.00 kg Then T = 0.795 s 14.82 EVALUATE: The rate at which the period changes is not constant but instead increases in time, even though the rate at which the water flows out is constant k IDENTIFY: Use Fx = − kx to determine k for the wire Then f = 2π m SET UP: F = mg moves the end of the wire a distance Δl © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion EXECUTE: The force constant for this wire is k = f = 14.83 2π k = m 2π g = Δl 2π 9.80m/s 2.00 ×10−3 m 14-25 mg , so Δl = 11.1 Hz EVALUATE: The frequency is independent of the additional distance the ball is pulled downward, so long as that distance is small IDENTIFY and SET UP: Measure x from the equilibrium position of the object, where the gravity and spring forces balance Let + x be downward (a) Use conservation of energy (Eq.14.21) to relate vx and x Use Eq (14.21) to relate T to k/m mv + kx = kA2 x 2 2 = 0, mvx = 12 kA2 and v = A EXECUTE: For x k/m : T = 2π m/k implies k /m , just as for horizontal SHM We can use the period to calculate k /m = 2π /T Thus v = 2π A/T = 2π (0.100 m)/4.20 s = 0.150 m/s (b) IDENTIFY and SET UP: Use Eq (14.4) to relate a x and x EXECUTE: ma x = − kx so a x = − (k /m) x +x-direction is downward, so here x = −0.050 m a x = − (2π /T )2 ( −0.050 m) = + (2π /4.20 s) (0.050 m) = 0.112 m/s (positive, so direction is downward) (c) IDENTIFY and SET UP: Use Eq (14.13) to relate x and t The time asked for is twice the time it takes to go from x = to x = +0.050 m EXECUTE: x(t ) = Acos(ωt + φ ) Let φ = −π /2 , so x = at t = Then x = Acos(ωt − π /2) = A sin ωt = Asin(2π t /T ) Find the time t that gives x = +0.050 m: 0.050 m = (0.100 m) sin(2π t /T ) 2π t /T = arcsin(0.50) = π /6 and t = T /12 = 4.20 s/12 = 0.350 s The time asked for in the problem is twice this, 0.700 s (d) IDENTIFY: The problem is asking for the distance d that the spring stretches when the object hangs at rest from it Apply Newton’s second law to the object SET UP: The free-body diagram for the object is given in Figure 14.83 EXECUTE: ∑ Fx = ma x mg − kd = d = ( m /k ) g Figure 14.83 But k /m = 2π /T (part (a)) and m /k = (T /2π ) 2 ⎛ T ⎞ ⎛ 4.20 s ⎞ d =⎜ ⎟ g =⎜ (9.80 m/s ) = 4.38 m ⎝ 2π ⎠ ⎝ 2π ⎟⎠ 14.84 EVALUATE: When the displacement is upward (part (b)), the acceleration is downward The mass of the partridge is never entered into the calculation We used just the ratio k/m, that is determined from T IDENTIFY: x (t ) = A cos(ωt + φ ), vx = − Aω sin(ωt + φ ) and a x = −ω x ω = 2π /T SET UP: x = A when t = gives φ = ⎛ 2π (0.240 m) ⎞ ⎛ 2π t ⎞ ⎛ 2π t ⎞ ⎛ 2π t ⎞ EXECUTE: x = (0.240 m)cos ⎜ v = −⎜ ⎟ sin ⎜ ⎟ = −(1.00530 m/s)sin ⎜ ⎟ ⎝ 1.50 s ⎟⎠ x ⎝ 1.50 s ⎠ ⎝ (1.50 s) ⎠ ⎝ 1.50 s ⎠ ⎛ 2π ⎞ ⎛ 2π t ⎞ ⎛ 2π t ⎞ ax = − ⎜ ⎟ (0.240 m)cos ⎜ ⎟ = −(4.2110 m/s )cos ⎜ ⎟ ⎝ 1.50 s ⎠ ⎝ 1.50 s ⎠ ⎝ 1.50 s ⎠ © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-26 Chapter 14 (a) Substitution gives x = − 0.120 m, or using t = T −A gives x = A cos 120° = (b) Substitution gives ma x = +(0.0200 kg)(2.106 m/s ) = 4.21 × 10−2 N, in the + x -direction T ⎛ −3 A/4 ⎞ arccos ⎜ ⎟ = 0.577 s 2π ⎝ A ⎠ (d) Using the time found in part (c), v = 0.665 m/s EVALUATE: We could also calculate the speed in part (d) from the conservation of energy expression, Eq (14.22) IDENTIFY: Apply conservation of linear momentum to the collision between the steak and the pan Then apply conservation of energy to the motion after the collision to find the amplitude of the subsequent SHM Use Eq (14.12) to calculate the period (a) SET UP: First find the speed of the steak just before it strikes the pan Use a coordinate system with + y downward (c) t = 14.85 v0 y = (released from the rest); y − y0 = 0.40 m; a y = + 9.80 m/s ; v y = ? v 2y = v02y + 2a y ( y − y0 ) EXECUTE: v y = + 2a y ( y − y0 ) = + 2(9.80 m/s )(0.40 m) = +2.80 m/s SET UP: Apply conservation of momentum to the collision between the steak and the pan After the collision the steak and the pan are moving together with common velocity v2 Let A be the steak and B be the pan The system before and after the collision is shown in Figure 14.85 Figure 14.85 EXECUTE: Py conserved: m Av A1 y + mB vB1 y = ( m A + mB )v2 y m Av A1 = (m A + mB )v2 ⎛ mA ⎞ ⎛ ⎞ 2.2 kg v2 = ⎜ v A1 = ⎜ (2.80 m/s) = 2.57 m/s ⎝ 2.2 kg + 0.20 kg ⎟⎠ ⎝ m A + mB ⎠⎟ (b) SET UP: Conservation of energy applied to the SHM gives: mv 2 + 12 kx02 = 12 kA2 where v0 and x0 are the initial speed and displacement of the object and where the displacement is measured from the equilibrium position of the object EXECUTE: The weight of the steak will stretch the spring an additional distance d given by kd = mg so mg (2.2 kg)(9.80 m/s ) = = 0.0539 m So just after the steak hits the pan, before the pan has had time 400 N/m k to move, the steak plus pan is 0.0539 m above the equilibrium position of the combined object Thus x0 = 0.0539 m From part (a) v0 = 2.57 m/s, the speed of the combined object just after the collision d= Then mv 2 + 12 kx02 = 12 kA2 gives A= (c) T = 2π m /k = 2π mv02 + kx02 2.4 kg(2.57 m/s) + (400 N/m)(0.0539 m)2 = = 0.21 m k 400 N/m 2.4 kg = 0.49 s 400 N/m © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14.86 14-27 EVALUATE: The amplitude is less than the initial height of the steak above the pan because mechanical energy is lost in the inelastic collision k Use energy considerations to find the new amplitude IDENTIFY: f = 2π m f = 0.600 Hz, m = 400 kg; f = SET UP: 2π k gives k = 5685 N/m This is the effective force constant m of the two springs (a) After the gravel sack falls off, the remaining mass attached to the springs is 225 kg The force constant of the springs is unaffected, so f = 0.800 Hz To find the new amplitude use energy considerations to find the distance downward that the beam travels after the gravel falls off Before the sack falls off, the amount x0 that the spring is stretched at equilibrium is given by mg − kx0 , so x0 = mg /k = (400 kg)(9.80 m/s )/(5685 N/m) = 0.6895 m The maximum upward displacement of the beam is A = 0.400 m above this point, so at this point the spring is stretched 0.2895 m With the new mass, the mass 225 kg of the beam alone, at equilibrium the spring is stretched mg/k = (225 kg)(9.80 m/s )/(5685 N/m) = 0.3879 m The new amplitude is therefore 0.3879 m − 0.2895 m = 0.098 m The beam moves 0.098 m above and below the new equilibrium position Energy calculations show that v = when the beam is 0.098 m above and below the equilibrium point (b) The remaining mass and the spring constant is the same in part (a), so the new frequency is again 0.800 Hz The sack falls off when the spring is stretched 0.6895 m And the speed of the beam at this point is v = A k /m = (5685 N/m)/(400 kg) = 1.508 m/s Take y = at this point The total energy of the beam at this point, just after the sack falls off, is E = K + U el + U grav = 12 (225 kg)(1.508 m/s)2 + (5685 N/m)(0.6895 m) + = 1608 J Let this be point Let point be where the beam has moved upward a distance d and where v = E2 = 12 k (0.6895 m − d ) + mgd E1 = E2 gives d = 0.7275 m At 14.87 this end point of motion the spring is compressed 0.7275 m – 0.6895 m = 0.0380 m At the new equilibrium position the spring is stretched 0.3879 m, so the new amplitude is 0.3879 m + 0.0380 m = 0.426 m Energy calculations show that v is also zero when the beam is 0.426 m below the equilibrium position EVALUATE: The new frequency is independent of the point in the motion at which the bag falls off The new amplitude is smaller than the original amplitude when the sack falls off at the maximum upward displacement of the beam The new amplitude is larger than the original amplitude when the sack falls off when the beam has maximum speed IDENTIFY and SET UP: Use Eq (14.12) to calculate g and use Eq (14.4) applied to Newtonia to relate g to the mass of the planet EXECUTE: The pendulum swings through 12 cycle in 1.42 s, so T = 2.84 s L = 1.85 m Use T to find g: T = 2π L/g so g = L(2π /T )2 = 9.055 m/s Use g to find the mass M p of Newtonia: g = GM p /Rp2 2π Rp = 5.14 × 107 m, so Rp = 8.18 × 106 m mp = 14.88 gRp2 G = 9.08 × 1024 kg EVALUATE: g is similar to that at the surface of the earth The radius of Newtonia is a little less than earth’s radius and its mass is a little more IDENTIFY: Fx = −kx allows us to calculate k T = 2π m /k x (t ) = A cos(ω t + φ ) Fnet = −kx SET UP: Let φ = π /2 so x (t ) = A sin(ωt ) At t = 0, x = and the object is moving downward When the object is below the equilibrium position, Fspring is upward EXECUTE: (a) Solving Eq (14.12) for m, and using k = F Δl © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-28 Chapter 14 2 ⎛ T ⎞ F ⎛ 1.00 s ⎞ 40.0 N m=⎜ =⎜ = 4.05 kg ⎟ ⎟ ⎝ 2π ⎠ Δl ⎝ 2π ⎠ 0.250 m (b) t = (0.35)T, and so x = − Asin[2π (0.35)] = −0.0405 m Since t > T/4, the mass has already passed the lowest point of its motion, and is on the way up (c) Taking upward forces to be positive, Fspring − mg = − kx, where x is the displacement from equilibrium, so Fspring = −(160 N/m)( −0.030 m) + (4.05 kg)(9.80 m/s ) = 44.5 N 14.89 EVALUATE: When the object is below the equilibrium position the net force is upward and the upward spring force is larger in magnitude than the downward weight of the object IDENTIFY: Use Eq (14.13) to relate x and t T = 3.5 s SET UP: The motion of the raft is sketched in Figure 14.89 Let the raft be at x = + A when t = Then φ = and x(t ) = A cos ωt Figure 14.89 EXECUTE: Calculate the time it takes the raft to move from x = + A = +0.200 m to x = A − 0.100 m = 0.100 m Write the equation for x(t) in terms of T rather than ω: ω = 2π /T gives that x(t ) = Acos(2π t /T ) x = A at t = x = 0.100 m implies 0.100 m = (0.200 m) cos(2π t /T ) cos (2π t /T ) = 0.500 so 2π t /T = arccos(0.500) = 1.047 rad t = (T /2π )(1.047 rad) = (3.5 s/2π )(1.047 rad) = 0.583 s This is the time for the raft to move down from x = 0.200 m to x = 0.100 m But people can also get off while the raft is moving up from x = 0.100 m to x = 0.200 m, so during each period of the motion the time the people have to get off is 2t = 2(0.583 s) = 1.17 s EVALUATE: The time to go from x = to x = A and return is T /2 = 1.75 s The time to go from x = A/2 14.90 to A and return is less than this IDENTIFY: T = 2π /ω Fr (r ) = − kr to determine k SET UP: Example 13.10 derives Fr ( r ) = − EXECUTE: 2π ω = 2π RE3 r ar = Fr /m is in the form of Eq (14.8), with x replaced by r, so the motion is simple harmonic k = T= GM E m GM E m RE3 ω2 = k GM E g = = The period is then m R RE E RE 6.38 × 106 m = 2π = 5070 s, or 84.5 g 9.80 m/s EVALUATE: The period is independent of the mass of the object but does depend on RE , which is also 14.91 the amplitude of the motion IDENTIFY: During the collision, linear momentum is conserved After the collision, mechanical energy is conserved and the motion is SHM SET UP: The linear momentum is px = mvx , the kinetic energy is 12 mv , and the potential energy is kx The period is T = 2π m , which is the target variable k © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14-29 EXECUTE: Apply conservation of linear momentum to the collision: (8.00 × 10−3 kg)(280 m/s) = (1.00 kg)v v = 2.24 m/s This is vmax for the SHM A = 0.180 m (given) So 2 ⎛v ⎞ ⎛ 2.24 m/s ⎞ mv = kA2 k = ⎜ max ⎟ m = ⎜ ⎟ (1.00 kg) = 154.9 N/m max A ⎝ ⎠ ⎝ 0.180 m ⎠ m 1.00 kg = 2π = 0.505 s 154.9 N/m k EVALUATE: This block would weigh about pounds, which is rather heavy, but the spring constant is large enough to keep the period within an easily observable range T = 2π 14.92 x IDENTIFY: U ( x) − U ( x0 ) = ∫ Fx dx In part (b) follow the steps outlined in the hint x0 SET UP: In part (a), let x0 = and U ( x0 ) = U (0) = The time for the object to go from x = to x = A is T/4 x x c EXECUTE: (a) U = − ∫ Fx dx = c ∫ x3dx = x 0 (b) From conservation of energy, mv x c dx = ( A4 − x ) vx = , so dt to A with respect to x and from to T /4 with respect to t, let u = root, A ∫0 dx A −x = c dt Integrating from 2m = c T To use the hint, 2m 4 dx A4 − x x , so that dx = A du and the upper limit of the u-integral is u = Factoring A2 out of the square A 7.41 m c 1 du 1.31 = = T , which may be expressed as T = ∫ A c A A 32m − u4 (c) The period does depend on amplitude, and the motion is not simple harmonic EVALUATE: Simple harmonic motion requires Fx = − kx, where k is a constant, and that is not the case 14.93 here IDENTIFY: Fr = − dU /dr The equilibrium separation req is given by F ( req ) = The force constant k is defined by Fr = − kx f = 2π k , where m is the reduced mass m SET UP: d (r − n )/dr = − nr − ( n +1) , for n ≥ EXECUTE: (a) Fr = − ⎡⎛ R ⎞ ⎤ dU = Α ⎢⎜ 90 ⎟ − ⎥ dr ⎢⎣⎝ r ⎠ r ⎥⎦ (b) Setting the above expression for Fr equal to zero, the term in square brackets vanishes, so that R07 req = req , or R07 = req , and req = R0 (c) U ( R0 ) = − 7A = −7.57 × 10−19 J R0 (d) The above expression for Fr can be expressed as Fr = Fr ≈ −9 −2 ⎛ r ⎞ ⎤ A A ⎡⎢⎛ r ⎞ −9 −2 − ⎜ ⎟ ⎜ ⎟ ⎥ = ⎡⎣(1 + ( x/R0 )) − (1 + ( x/R0 )) ⎤⎦ R ⎥ R02 ⎢⎝ R0 ⎠ R ⎝ 0⎠ ⎦ ⎣ A R02 [(1 − 9( x /R0 )) − (1 − 2( x /R0 ))] = A R02 ⎛ 7A ⎞ (− x /R0 ) = − ⎜ ⎟ x ⎜R ⎟ ⎝ 0⎠ © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-30 Chapter 14 (e) f = 1 k/m = 2π 2π 7A R03m = 8.39 × 1012 Hz EVALUATE: The force constant depends on the parameters A and R0 in the expression for U (r ) The minus sign in the expression in part (d) shows that for small displacements from equilibrium, Fr is a 14.94 14.95 restoring force IDENTIFY: Newton’s second law, in both its linear and rotational form, applies to this system The motion is SHM SET UP: ∑ F = macm and ∑τ = Iα , where I = MR for a solid sphere, and Rα = acm with no slipping ⎛2 ⎞ EXECUTE: For each sphere, fs R = ⎜ MR ⎟ α Rα = acm fs = Macm For the system of two spheres, ⎝5 ⎠ fs − kx = −2Macm 14 5⎛ k Macm − kx = −2Macm kx = Macm and acm = ⎜ 5 14 ⎝ M a x = −ω x so ω = 5k 2π 14 M 14(0.800 kg) T= = 2π = 2π = 0.743 s 14 M 5k 5(160 N/m) ω 5⎛ k ⎞ ⎟ x a x = − ⎜ 14 ⎠ ⎝M ⎞ ⎟ x ⎠ EVALUATE: If the surface were smooth, there would be no rolling, but the presence of friction provides the torque to cause the spheres to rotate IDENTIFY: Apply conservation of energy to the motion before and after the collision Apply conservation of linear momentum to the collision After the collision the system moves as a simple pendulum If the g maximum angular displacement is small, f = 2π L SET UP: In the motion before and after the collision there is energy conversion between gravitational potential energy mgh, where h is the height above the lowest point in the motion, and kinetic energy EXECUTE: Energy conservation during downward swing: m2 gh0 = 12 m2v and v = gh0 = 2(9.8 m/s )(0.100 m) = 1.40 m/s Momentum conservation during collision: m2v = ( m2 + m3 )V and V= m2v (2.00 kg)(1.40 m/s) = = 0.560 m/s m2 + m3 5.00 kg Energy conservation during upward swing: Mghf = hf = V /2 g = (0.560 m/s) 2(9.80 m/s ) MV and = 0.0160 m = 1.60 cm Figure 14.95 shows how the maximum angular displacement is calculated from hf cosθ = 48.4 cm and 50.0 cm g 9.80 m/s = = 0.705 Hz 2π l 2π 0.500 m EVALUATE: 14.5° = 0.253 rad sin(0.253 rad) = 0.250 sin θ ≈ θ and Eq (14.34) is accurate θ = 14.5° f = Figure 14.95 © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14.96 14-31 IDENTIFY: T = 2π I /mgd SET UP: The model for the leg is sketched in Figure 14.96 T = 2π I /mgd , m = 3M d = ycg = m1 y1 + m2 y2 For a rod with the axis at one end, I = 13 ML2 For a rod with the axis at its center, m1 + m2 ML2 I = 12 EXECUTE: d = M ([1.55 m]/2) + M (1.55 m + (1.55 m)/2) = 1.292 m I + I1 + I 3M I1 = 13 (2M )(1.55 m) = (1.602 m ) M I 2,cm = 12 M (1.55 m) The parallel-axis theorem (Eq 9.19) gives I = I 2,cm + M (1.55 m + [1.55 m]/2) = (5.606 m ) M I = I1 + I = (7.208 m ) M Then T = 2π I /mgd = 2π (7.208 m ) M (3M )(9.80 m/s )(1.292 m) = 2.74 s EVALUATE: This is a little smaller than T = 2.9 s found in Example 14.10 Figure 14.96 14.97 IDENTIFY: The motion is simple harmonic if the equation of motion for the angular oscillations is of the d 2θ κ form = − θ , and in this case the period is T = 2π I/κ I dt SET UP: For a slender rod pivoted about its center, I = 12 ML2 d 2θ ⎛ L ⎞L EXECUTE: The torque on the rod about the pivot is τ = − ⎜ k θ ⎟ τ = Iα = I gives ⎝ ⎠2 dt d 2θ dt =−k T = 2π 14.98 κ 3k L2 /4 3k d 2θ , is proportional to θ and the motion is angular SHM = θ = − θ I M I M dt M 3k ⎛ L ⎞L EVALUATE: The expression we used for the torque, τ = - ⎜ k θ ⎟ , is valid only when θ is small ⎝ ⎠2 enough for sin θ ≈ θ and cosθ ≈ IDENTIFY and SET UP: Eq (14.39) gives the period for the bell and Eq (14.34) gives the period for the clapper EXECUTE: The bell swings as a physical pendulum so its period of oscillation is given by T = 2π I/mgd = 2π 18.0 kg ⋅ m /(34.0 kg)(9.80 m/s )(0.60 m) = 1.885 s The clapper is a simple pendulum so its period is given by T = 2π L /g Thus L = g (T /2π ) = (9.80 m/s )(1.885 s/2π ) = 0.88 m EVALUATE: If the cm of the bell were at the geometrical center of the bell, the bell would extend 1.20 m from the pivot, so the clapper is well inside tbe bell © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 14-32 14.99 Chapter 14 IDENTIFY: The object oscillates as a physical pendulum, with f = 2π mgd , where m is the total mass I of the object SET UP: The moment of inertia about the pivot is 2(1/3) ML2 = (2/3) ML2 , and the center of gravity when balanced is a distance d = L /(2 2) below the pivot EXECUTE: The frequency is f = EVALUATE: If fsp = 6g = L 4π 6g 2L g is the frequency for a simple pendulum of length L, L fsp = 1.03 fsp IDENTIFY: The angular frequency is given by Eq (14.38) Use the parallel-axis theorem to calculate I in terms of x (a) SET UP: f = 14.100 2π 1 = T 2π Figure 14.100 d = x, the distance from the cg of the object (which is at its geometrical center) to the pivot EXECUTE: I is the moment of inertia about the axis of rotation through O By the parallel axis theorem mgx gx mL2 (Table 9.2), so I = mx + mL2 ω = I = md + I cm I cm = 12 = 12 2 mx + 12 mL x + L2 /12 (b) The maximum ω as x varies occurs when d ω /dx = x −1/2 2 1/ ( x + L /12) x −1/2 − − dω = gives dx g ⎞ d ⎛ x1/ ⎜ ⎟ = 2 1/2 dx ⎜⎝ ( x + L /12) ⎟⎠ 2x ( x1/2 ) = 2 ( x + L2 /12)3/2 x3/2 x + L2 /12 =0 x + L2 /12 = x so x = L / 12 Get maximum ω when the pivot is a distance L / 12 above the center of the rod (c) To answer this question we need an expression for ωmax : In ω = ωmax = gx x + L2 /12 substitute x = L / 12 g ( L/ 12) L2 /12 + L2 /12 = g1/2 (12) −1/ ( L/6)1/2 = g/L (12) −1/ (6)1/2 = g/L (3)1/4 2 ωmax = ( g /L) and L = g 3/ωmax ωmax = 2π rad/s gives L = (9.80 m/s ) (2π rad/s) = 0.430 m © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Periodic Motion 14-33 EVALUATE: ω → as x → and ω → g /(2 L) = 1.225 g /L when x → L /2 ωmax is greater than the x = L /2 value A simple pendulum has ω = g /L ; ωmax is greater than this 14.101 IDENTIFY: In each situation, imagine the mass moves a distance Δ x, the springs move distances Δ x1 and Δ x2 , with forces F1 = − k1Δx1, F2 = − k2Δx2 SET UP: Let Δ x1 and Δ x2 be positive if the springs are stretched, negative if compressed EXECUTE: (a) Δ x = Δ x1 = Δ x2 , F = F1 + F2 = −(k1 + k2 )Δ x, so keff = k1 + k2 (b) Despite the orientation of the springs, and the fact that one will be compressed when the other is extended, Δ x = Δ x1 − Δ x2 and both spring forces are in the same direction The above result is still valid; keff = k1 + k2 (c) For massless springs, the force on the block must be equal to the tension in any point of the spring ⎛1 F F ⎞ k +k combination, and F = F1 = F2 Δ x1 = − , Δ x2 == − , Δ x = − ⎜ + ⎟ F = − F and k1 k2 k k k1k2 2⎠ ⎝ k1k2 k1 + k2 (d) The result of part (c) shows that when a spring is cut in half, the effective spring constant doubles, and so the frequency increases by a factor of EVALUATE: In cases (a) and (b) the effective force constant is greater than either k1 or k2 and in case (c) keff = 14.102 it is less IDENTIFY: Calculate Fnet and define keff by Fnet = − keff x T = 2π m /keff SET UP: If the elongations of the springs are x1 and x2 , they must satisfy x1 + x2 = 0.200 m EXECUTE: (a) The net force on the block at equilibrium is zero, and so k1x1 = k2 x2 and one spring (the one with k1 = 2.00 N/m) must be stretched three times as much as the one with k2 = 6.00 Ν /m The sum of the elongations is 0.200 m, and so one spring stretches 0.150 m and the other stretches 0.050 m, and so the equilibrium lengths are 0.350 m and 0.250 m (b) When the block is displaced a distance x to the right, the net force on the block is − k1 ( x1 + x) + k2 ( x2 − x) = −[k1x1 − k2 x2 ] − (k1 + k2 ) x From the result of part (a), the term in square brackets is zero, and so the net force is − (k1 + k2 ) x, the effective spring constant is keff = k1 + k2 and the period of vibration is T = 2π 0.100 kg = 0.702 s 8.00 N/m EVALUATE: The motion is the same as if the block were attached to a single spring that has force constant keff 14.103 IDENTIFY: Follow the procedure specified in the hint SET UP: Denote the position of a piece of the spring by l ; l = is the fixed point and l = L is the moving end of the spring Then the velocity of the point corresponding to l , denoted u, is u (l ) = v l L (when the spring is moving, l will be a function of time, and so u is an implicit function of time) EXECUTE: (a) dm = (b) mv M Mv 1 Mv 2 dl , and so dK = dm u = l dl and K = ∫ dK = L 2 L L3 L ∫0 l dl = Mv dv dx + kx = 0, or ma + kx = 0, which is Eq (14.4) dt dt M 3k M , so ω = and M ′ = 3 M EVALUATE: The effective mass of the spring is only one-third of its actual mass (c) m is replaced by © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher