Work – engergy and impilse – momentum principles

Chapter 4. Work – Engergy and Impilse – Momentum Principles for a Particle A. Work – Engergy Principle 4.1. (a) Compute the work done by each force given in the following list as its point of application moves from 1 to 3 along the straight line connecting 1 and 3. (b) Repeat part (a) if the path consists of the straight line segments 1-2 and 2-3. (x and y are in m) 1. F=30i-10j N 2. F=3xi-yj N 3. F=3yi-xj N 4.2. Compute the work of the force F=(F 0 /b 3 )(xy 2 i +x 2 yj) as its point of application moves from 1 to 2 along (a) the line y=x; and (b) the parabola y=x 2 /b. 4.3 Repeat Prob. 4.2 for the force F = (F 0 /b 3 )(x 2 yi+ x y 2 j). 4.4 The collar of weight W slides on a frictionless circular arc of radius R. The ideal spring attached to the collar has the free length L 0 = R and stiffness k. When the slider moves from A to B, compute (a) the work done by the spring; and (b) the work done by the weight. 4.5 Derive the expression for the work done by the ideal spring on the slider when the slider moves from A to B. Assume that the free length of the spring is (a) L 0 = b; and (b) L 0 = 0.8b. 4.6 The man slides the 100-kg crate across the floor by pulling with a constant force of 200 N. If the crate was initially at rest, how far will the crate move before its speed is 1 m/s? The coefficient of kinetic friction between the crate and the floor is 0.18. 4.7 The 5-lb package arrives at A, the top of the inclined roller conveyor, with a speed of 5 m/s. After descending the conveyor, the package slides a dis- tance d on the rough horizontal surface, coming to a stop at B. If the coefficient of kinetic friction between the package and the horizontal plane is 0.4, determine the distance d. 4.8 The speed of the car at the base of a 10-m hill is 54 km/h. Assuming the driver keeps her foot off the brake and accelerator pedals, what will be the speed of the car at the top of the hill? 4.9 The 0.8-kg slider is at rest in position 1 when the constant vertical force F is applied to the rope that is attached to the slider. What is the required magnitude of F if the slider is to reach position 2 with a speed of 6 m/s? Neglect friction. 4.10 A crate of weight W is dragged across the floor from A to B by the constant vertical force P acting at the end of the rope. Calculate the work done on the crate by the force P. Assume that the crate does not lift off the floor. 4.11 The 0.31-kg mass slides on a frictionless wire that lies in the vertical plane. The ideal spring attached to the mass has a free length of 80 mm and its stiffness is 120 N/m. Calculate the smallest value of the distance b if the mass is to reach the end of the wire at B after being released from rest at A. 4.12 The 1-kg collar moves from A to B along a frictionless rod. The stiffness of the spring is k and its free length is 200 mm. Compute the value of k so that the slider arrives at B with a speed of 1m/s after being released from rest at A. 4.13 The 2-kg weight is released from rest in position A, where the two springs of stiffness k each are undeformed. Determine the largest k for which the weight would reach position B. 4.14 The sliding collar of weight W = 10 N is attached to two springs of stiff- nesses k 1 = 180 N/m and k 2 = 60 N/m. The free length of each spring is 50 cm. If the collar is released from rest in position A, determine its speed in position B. Neglect friction. 4.15 The spring attached to the 0.6-kg sliding collar has a stiffness of 200 N/m and a free length of 150 mm. If the speed of the collar in position A is 3 m/s to the right, determine the speed in position B. Neglect friction. 4.16 The 0.5-kg pendulum oscillates with the amplitude of  max = 50 ◦ . Determine the maximum force in the supporting string. 4.17 The semicircular rod AC lies in the vertical plane. The spring wound around the rod is undeformed when  = 45 ◦ . If the 210-g slider is pressed against the spring and released at  = 30 ◦ , determine the velocity of the slider when it passes through B. Neglect friction and assume the slider is not attached to the spring. B. Impulse – Momentum Principle 4.18 The velocity of a 2-kg particle at time t = 0 is v = 10i m/s. Determine the velocity at t = 5 s if the particle is acted upon by the force F = 2ti 0.6t 2 j N, where t is in seconds. 4.19 A constant horizontal force P (not shown) acts on the 0.5 kg body as it slides along a frictionless, horizontal table. During the time interval t = 5 s to t = 7.5 s, the velocity changes as indicated in the figure. Determine the magnitude and direction of P. 4.20 The 0.2-kg mass moves in the vertical x y-plane. At time t= 0, the velocity of the mass is 8j m/s. In addition to its weight, the mass is acted on by the force F(t) = F(t)i, where the magnitude of the force varies with time as shown in the figure. Determine the velocity vector of the mass at t = 4 s. 4.21 The 60-kg crate is sliding down the inclined plane. The coefficient of kinetic friction between the crate and the plane is 0.2, and the force P applied to the crate is constant. If the speed of the crate changes from 8 m/s to zero in 3 seconds, determine P. 4.22 A parcel is lowered onto a conveyor belt that is moving at 4 m/s. If the coefficient of kinetic friction between the parcel and the belt is 0.25, calculate the time that it takes for the parcel to reach the speed of the belt. 4.23 The force P 0 has constant magnitude and direction, but its point of application C moves along the x-axis with the constant speed v 0 . Determine the angular impulse of the force about point A for the time period during which C moves from A to B. 4.24 The velocity of the 500-g particle at B is v=2i+4j+6k m/s. Calculate the angular momentum of the particle about point A at this instant. 4.25 The particle of mass m is restrained by a string to travel around a circular path on the horizontal table. The coefficient of kinetic friction between the particle and the table is 0.15. If the initial speed of the particle is v 1 = 8 m/s, determine the time that elapses before the particle stops. 4.26 A rope wound around the rim of the wheel is pulled by force F, which varies with time as shown. Calculate the angular impulse of F about the center of the wheel during the time interval t = 0 to t = 0.5 s. 4.27 The 0.6-kg mass is supported by two arms of negligible mass. The angle  of the arms can be varied by changing the force F acting on the sliding collar. When  = 70 ◦ , the assembly is rotating freely about the vertical axis with the angular velocity  = 15 rad/s. Determine the angular velocity after  is reduced to 30 ◦ . . Chapter 4. Work – Engergy and Impilse – Momentum Principles for a Particle A. Work – Engergy Principle 4.1. (a) Compute the work done by each force given in the. length L 0 = R and stiffness k. When the slider moves from A to B, compute (a) the work done by the spring; and (b) the work done by the weight. 4.5 Derive the expression for the work done by. connecting 1 and 3. (b) Repeat part (a) if the path consists of the straight line segments 1-2 and 2-3. (x and y are in m) 1. F=30i-10j N 2. F=3xi-yj N 3. F=3yi-xj N 4.2. Compute the work of