De thi vat li quoc te IPHO nam 20036

When the weight has fallen a distance D and stopped, the law of conservation of total mechanical energy as applied to the particle-weight pair as a system leads to.. −Mgh =E ' − Mg (h+[r]

Solution- Theoretical Question 1 A Swing with a Falling Weight Part A

(a) Since the length of the string L=s+Rθ is constant, its rate of change must be zero Hence we have

˙s + R ˙θ=0 (A1)*1

(b) Relative to O, Q moves on a circle of radius R with angular velocity θ˙ , so ⃗

vQ=R ˙θ ^t=− ˙s ^t (A2)* (c) Refer to Fig A1 Relative to Q, the displacement of P in a time interval t is

Δ ⃗r'

=(sΔθ)(− ^r)+(Δs)^t=[(s ˙θ)(− ^r )+˙s ^t] Δt It follows

⃗

v'=− s ˙θ ^r+ ˙s ^t (A3)*

(d) The velocity of the particle relative to O is the sum of the two relative velocities given in Eqs (A2) and (A3) so that

⃗ v =⃗v'

+⃗vQ=(− s ˙θ ^r+ ˙s ^t )+R ˙θ ^t=− s ˙θ ^r (A4)*

(e) Refer to Fig A2 The ( − ^t )-component of the velocity change Δ ⃗v is given by (− ^t )⋅ Δ ⃗v=vΔθ=v ˙θ Δt Therefore, the t^ -component of the

acceleration ⃗a=Δ ⃗v / Δt is given by t^⋅ ^a=− v ˙θ Since the speed v of the particle is s ˙θ according to Eq (A4), we see that the t^ -component of the particle’s acceleration at P is given by

⃗

a⋅ ^t=− v ˙θ=−(s ˙θ) ˙θ=− s ˙θ2

(A5)*

Note that, from Fig A2, the radial component of the acceleration may also be obtained as ⃗a⋅ ^r=−dv /dt=−ds ˙θ¿/dt ¿

1 An equation marked with an asterisk contains answer to the problem.

tˆ



Q

rˆ 

v⃗



P

 

v

Figure A2

O 

v⃗

 v v

⃗ ⃗

 

v v

tˆ

Q

rˆ 

s

s+s 

s P Figure A1

r ⃗

(f) Refer to Fig A3 The gravitational potential energy of the particle is given by U=− mgh It may be expressed in terms of s and  as

U (θ)=− mg[ R(1 −cosθ)+s sin θ] (A6)*

(g) At the lowest point of its trajectory, the particle’s gravitational potential energy U must assume its minimum value Um If the particle’s mechanical energy E were equal to Um, its kinetic energy would be zero The particle would then remain stationary and be in the static equilibrium state shown in Fig A4 Thus, the potential energy reaches its minimum value when  =  /2 or s = L−  R /2.

From Fig A4 or Eq (A6), the minimum potential energy is then Um=U (π

2)=− mg[R+L −(πR/2)] (A7) Initially, the total mechanical energy E is Since E is conserved, the speed vm of the particle at the lowest point of its trajectory must satisfy

E=0=1 2mvm

2

+Um (A8) From Eqs (A7) and (A8), we obtain

vm=√− 2Um/m=√2 g [R+(L − πR/2)] (A9)* Part B

(h) From Eq (A6), the total mechanical energy of the particle may be written as R

x

O s

A Q

s sin

R−R cos

P

R cos Figure A3 h

R x

O s

A

Q

P (at rest)

Figure A4

E=0=1 2mv

2

+U (θ)=1 2mv

2

− mg[R (1− cos θ)+s sin θ] (B1) From Eq (A4), the speed v is equal to s ˙θ Therefore, Eq (B1) implies

s ˙θ¿2=2 g [R (1− cos θ)+ssin θ]

v2=¿ (B2)

Let T be the tension in the string Then, as Fig B1 shows, the t^ -component of the net force on the particle is –T + mg sin  From Eq (A5), the tangential acceleration of the particle is (− s ˙θ2

) Thus, by Newton’s second law, we have

m(− s ˙θ2)=−T +mg sin θ (B3)

According to the last two equations, the tension may be expressed as T =m(s ˙θ2

+g sin θ)=mg

s [2 R(1− cos θ)+3 s sin θ] mgR

s [tan θ 2−

3 2(θ−

L

R)](sin θ) mgR

s (y1− y2)(sin θ)

(B4)

The functions y1=tan(θ/2) and

θ − L/ R¿/2

y2=3¿ are plotted in Fig B2

From Eq (B4) and Fig B2, we obtain the result shown in Table B1 The angle at R



x

O s

A Q

P mg Figure B1

T mg sin





Figure B2 y

30 20

y1=tanθ y1=tanθ

2 10

θ θs

0 -10

y2=3 2(θ −

L R) -20

0  3  / 2  3

-30

which y2 = y1 is called θs ( π <θs<2 π ) and is given by

2(θs−

L R)=tan

θs

2 (B5) or, equivalently, by

L R=θs−

2 3tan

θs

2 (B6) Since the ratio L/R is known to be given by

L R=

9 π +

2 3cot

π

16=(π + π 8)−

2 3tan

1 2(π +

π

8) (B7) one can readily see from the last two equations that θs=9 π /8

Table B1 shows that the tension T must be positive (or the string must be taut and straight) in the angular range 0<  <s Once  reaches s, the tension T becomes zero and the part of the string not in contact with the rod will not be straight afterwards The shortest possible value smin for the length s of the line

segment QP therefore occurs at θ=θs and is given by

smin=L− Rθs=R(9 π +

2 3cot

π 16 −

9 π )=

2 R cot

π

16=3 352 R (B8) When θ=θs , we have T = and Eqs (B2) and (B3) then leads to

v2=− gs sin θ Hence the speed vs is

vs=√−gsminsin θs=√2 gR

3 cot

π

16 sin

π

8=√

4 gR

3 cos

π

16

1 133√gR (B9)*

(i) When θ ≥θs , the particle moves like a projectile under gravity As shown in Fig B3, it is projected with an initial speed vs from the position P=(xs, ys)

in a direction making an angle φ=(3 π /2 −θs) with the y-axis.

The speed vH of the particle at the highest point of its parabolic trajectory

is equal to the y-component of its initial velocity when projected Thus,

vH=vssin (θs− π )=√4 gR3 cos16π sin π8=0 4334√gR (B10)*

The horizontal distance H traveled by the particle from point P to the point of Table B1

(y1− y2) sin θ tension T

0<θ<π positive positive positive

θ=π +  positive

π <θ<θs negative negative positive

maximum height is

H=vs

2

sin 2(θs− π )

2 g =

vs

2 2 gsin

9 π

4 =0 4535 R (B11)

The coordinates of the particle when θ=θs are given by xs=R cos θs− sminsin θs=− R cosπ

8+sminsin π

8=0 358 R (B12) ys=R sin θs+smincos θs=− R sin π8− smincosπ8=− 478 R (B13) Evidently, we have ¿ys∨¿(R+H ) Therefore the particle can indeed reach its

maximum height without striking the surface of the rod Part C

(j) Assume the weight is initially lower than O by h as shown in Fig C1.

When the weight has fallen a distance D and stopped, the law of conservation of total mechanical energy as applied to the particle-weight pair as a system leads to

−Mgh =E'− Mg (h+D) (C1) where E is the total mechanical energy of the particle when the weight has stopped It follows

E'=MgD

(C2) Let  be the total length of the string Then, its value at  = must be the same

A

h

M m

L

R

Figure C1

x



O m

R L

s

min

s

x

O

 s 

s

v

H

v

s



Figure B3

y

Q



) , (xs ys P 

as at any other angular displacement  Thus we must have Λ=L+π

2 R+h=s+R(θ + π

2)+(h+D) (C3) Noting that D =  L and introducing ℓ = L−D, we may write

ℓ=L − D=(1− α) L (C4) From the last two equations, we obtain

s=L − D − Rθ=ℓ − Rθ (C5) After the weight has stopped, the total mechanical energy of the particle must be conserved According to Eq (C2), we now have, instead of Eq (B1), the following equation:

E'=MgD=1 2mv

2

−mg [R(1 − cos θ)+s sin θ] (C6) The square of the particle’s speed is accordingly given by

s ˙θ¿2=2 MgD

m +2 gR[(1 − cos θ)+ s Rsinθ ] v2=¿

(C7) Since Eq (B3) stills applies, the tension T of the string is given by

−T +mg sin θ=m(− s ˙θ2) (C8) From the last two equations, it follows

T =m(s ˙θ2+g sin θ) mg

s [ 2 M

m D+2 R(1 −cos θ)+3 s sin θ] mgR

s [ MD

mR +(1− cos θ)+ 2(

ℓ

R−θ)sin θ]

(C9)

where Eq (C5) has been used to obtain the last equality We now introduce the function

f (θ)=1− cos θ+3 2(

ℓ

R−θ)sin θ (C10) From the fact ℓ = (L−D) >> R, we may write

f (θ)≈ 1+3

ℓ

Rsin θ −cos θ=1+ A sin(θ − φ) (C11) where we have introduced

3

ℓ R¿

2

1+¿

A=√¿

,

3 ℓ 2 R¿

2

¿

1+¿

√¿

φ=tan− 1 3 ℓ 2 R

¿

(C12)

3

ℓ R¿

2 1+¿

fmin=1 − A=1 −√¿

(C13)

Since the tension T remains nonnegative as the particle swings around the rod, we have from Eq (C9) the inequality

3 ℓ 2 R¿

2

¿

1+¿

MD

mR +fmin=

M (L − ℓ)

mR +1 −√¿

(C14)

or

3 ℓ 2 R¿

2

¿

1+¿

(ML

mR)+1≥ ( Mℓ mR )+√¿

(C15)

From Eq (C4), Eq (C15) may be written as (ML

mR)+1≥ [( ML mR )+(

3 L

2 R)](1 − α) (C16) Neglecting terms of the order (R/L) or higher, the last inequality leads to

α ≥ 1−

(ML mR)+1 (ML

mR)+( 3 L 2 R)

= (3 L

2 R)−1 (ML

mR)+( 3 L 2 R)

=

1−2 R 3 L 2 M

3 m+1

≈

1+2 M 3 m

(C17)

The critical value for the ratio D/L is therefore αc=

1 (1+2 M

3 m)