VECTOR MECH...,STAT.+DYN.(LL)-W/ACCESS
VECTOR MECH...,STAT.+DYN.(LL)-W/ACCESS
12th Edition
ISBN: 9781260265453
Author: BEER
Publisher: MCG
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Textbook Question
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Chapter 17.3, Problem 17.111P

A uniform slender rod of length L is dropped onto rigid supports at A and B. Because support B is slightly lower than support A, the rod strikes A with a velocity v ¯ 1 before it strikes B. Assuming perfectly elastic impact at both A and B, determine the angular velocity of the rod and the velocity of its mass center immediately after the rod (a) strikes support A, (b) strikes support B, (c) again strikes support A.

Chapter 17.3, Problem 17.111P, A uniform slender rod of length L is dropped onto rigid supports at A and B. Because support B is

Fig. P17.111

(a)

Expert Solution
Check Mark
To determine

Find the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support A.

Answer to Problem 17.111P

The angular velocity of the rod when mass center immediately after the rod strikes support A is 3v1L_.

The velocity of the rod when mass center immediately after the rod strikes support A 12v1_.

Explanation of Solution

Given information:

The length of uniform slender rod is L.

The mass of uniform slender rod is m.

The initial velocity of bar before it strikes B is v¯1.

Calculation:

Write the equation of centroidal moment of inertia (I¯) of the uniform slender rod.

I¯=112mL2

The impact is perfectly elastic at both A and B. Therefore, the coefficient of restitution is one (e=1).

Write the impact condition for the given system after the rod strikes support A.

(vA)2=v1

Here, (vA)2 is the velocity of the uniform slender rod after the rod strikes support A.

Write the equation of velocity of rod (v¯2) at its mass center after the rod strikes support A.

v¯2=L2ω2(vA)2

Here, ω2 is the angular velocity of the uniform slender rod after rod strikes support A.

Substitute v1 for (vA)2.

v¯2=L2ω2v1 (1)

Consider the impulse and momentum principle.

Sketch the impulse and momentum diagram for the first impact at A of the bar as shown in Figure (1).

VECTOR MECH...,STAT.+DYN.(LL)-W/ACCESS, Chapter 17.3, Problem 17.111P , additional homework tip  1

Here, Adt is the magnitude of the impulse acting at support A and dt is the time interval.

Refer Figure (1).

Take moment about A (positive sign in clockwise direction).

mv1L2+0=mv¯2L2+I¯ω2mv1L2=mv¯2L2+I¯ω2

Substitute 112mL2 for I¯ and L2ω2v1 for v¯2

mv1L2=m(L2ω2v1)L2+(112mL2)ω2=mL24ω2mv1L2+112mL2ω2

Simplify the Equation:

mv1L2+mv1L2=14mL2ω2+112mL2ω222mv1L=412mL2ω2mv1L=13mL2ω2ω2=3mv1LmL2ω2=3v1L

Thus, the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support A is 3v1L_.

Find the velocity of the rod when mass center immediately after the rod strikes support A using Equation (1).

v¯2=L2ω2v1

Substitute 3v1L for ω2.

v¯2=L2(3v1L)v1=3v12v1=12v1

Thus, the velocity of the rod when mass center immediately after the rod strikes support A 12v1_.

(b)

Expert Solution
Check Mark
To determine

Find the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support B.

Answer to Problem 17.111P

The angular velocity of the rod when mass center immediately after the rod strikes support B is 3v1L_.

The velocity of the rod when mass center immediately after the rod strikes support B is 12v1_.

Explanation of Solution

Calculation:

Write the impact condition for the given system after the rod strikes support B.

(vB)3=2v1

Here, (vA)2 is the velocity of the uniform slender rod after the rod strikes support B.

Write the equation of velocity of rod (v¯3) at its mass center after the rod strikes support B.

v¯3=(vB)3L2ω3

Here, ω3 is the angular velocity of the uniform slender rod after rod strikes support B.

Substitute 2v1 for (vB)3.

v¯3=2v1L2ω3 (2)

Consider the impulse and momentum principle.

Sketch the impulse and momentum diagram for impact at B of the bar as shown in Figure (2).

VECTOR MECH...,STAT.+DYN.(LL)-W/ACCESS, Chapter 17.3, Problem 17.111P , additional homework tip  2

Here, Bdt is the magnitude of the impulse acting at B.

Refer Figure (2).

Take moment about B (positive sign in clockwise direction).

mv¯2L2+I¯ω2+0=mv¯3L2I¯ω3mv¯2L2+I¯ω2=mv¯3L2I¯ω3

Substitute 112mL2 for I¯, 12v1 for v¯2, 3v1L for ω2, and 2v1L2ω3 for v¯3.

m(12v1)L2+(112mL2)3v1L=m(2v1L2ω3)L2(112mL2)ω314mv1L+14mv1L=2mv1L2mL24ω3112mL2ω30=2mv1L2mL24ω3112mL2ω3

2mv1L2=mL24ω3+112mL2ω3mv1L=13mL2ω3ω3=3mv1LmL2ω3=3v1LL

Thus, the angular velocity of the rod when mass center immediately after the rod strikes support B is 3v1L_.

Find the velocity of the rod when mass center immediately after the rod strikes support B using Equation (2).

v¯3=2v1L2ω3

Substitute 3v1L for ω3.

v¯3=2v1L2(3v1L)=2v13v12=12v1

Thus, the velocity of the rod when mass center immediately after the rod strikes support B 12v1_.

(c)

Expert Solution
Check Mark
To determine

Find the angular velocity of the rod and the velocity of its mass center immediately after the rod strikes support A again.

Answer to Problem 17.111P

The angular velocity of the rod when mass center immediately after the rod strikes support A again is 0_.

The velocity of the rod when mass center immediately after the rod strikes support A again is v1_.

Explanation of Solution

Calculation:

Write the impact condition for the given system after the rod again strikes support A.

(vA)4=v1

Here, (vA)4 is the velocity of the uniform slender rod the rod again strikes support A.

Write the equation of velocity of rod (v¯4) at its mass center after the rod again strikes support.

v¯4=(vA)4+L2ω4

Here, ω4 is the angular velocity of the uniform slender rod after rod again strikes support A.

Substitute v1 for (vA)4.

v¯4=v1+L2ω4 (3)

Consider the impulse and momentum principle.

Sketch the impulse and momentum diagram for second impact at A of the bar as shown in Figure (3).

VECTOR MECH...,STAT.+DYN.(LL)-W/ACCESS, Chapter 17.3, Problem 17.111P , additional homework tip  3

Refer Figure (3).

Take moment about A (positive sign in clockwise direction).

mv¯3L2+I¯ω3+0=mv¯4L2+I¯ω4mv¯3L2+I¯ω3=mv¯4L2+I¯ω4

Substitute 112mL2 for I¯, 12v1 for v¯3, 3v1L for ω3, and v1+L2ω4 for v¯4.

m(12v1)L2+(112mL2)(3v1L)=m(v1+L2ω4)L2+(112mL2)ω414mv1L+14mv1L=12mv1L+mL24ω4+112mL2ω414mv1L+14mv1L12mv1L=mL24ω4+112mL2ω40=14mL2ω4ω4=0

Thus, the angular velocity of the uniform slender rod after rod again strikes support A support B is 0_.

Find the velocity of rod at its mass center after the rod again strikes support A using Equation (3).

v¯4=v1+L2ω4

Substitute 0 for ω4.

v¯4=v1+0=v1

Thus, the velocity of rod at its mass center after the rod again strikes support A is v1_.

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Chapter 17 Solutions

VECTOR MECH...,STAT.+DYN.(LL)-W/ACCESS

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