(a)
Find the coefficient of restitution between A and B
(a)
Answer to Problem 13.178P
The coefficient of restitution between A and B
Explanation of Solution
Given information:
The weight of the block A
The weight of the block B
The weight of the block C
The coefficient of friction between the block and plane
The initial speed of the block A
The blocks B and C are at rest.
The distance between the blocks (d) is
The width of the each blocks (b) is
The acceleration due to gravity (g) is
Calculation:
Calculate the mass of the block A
Substitute
Calculate the mass of the block B
Substitute
Calculate the mass of the block C
Substitute
Show the diagram of the block A just before its impact with block B as in Figure (1).
The expression for the initial kinetic energy of the block A at position ‘1’
Here,
The expression for the kinetic energy of the block A at position ‘2’ just before its impact with blocks B
Here,
The expression for the work done by the block A to overcome frictional force
The expression for the principle of work and energy to the block A at position ‘1’ and position ‘2’ just before its impact with block B as follows:
Substitute
Substitute
Show the diagram of the block A just after its impact with block B as in Figure (2).
The expression for the kinetic energy of the block A immediately after the impact
Here,
The block finally comes to stop after the impact. Thus,
The expression for the work done by the block A after the collision to overcome the frictional force
The expression for the principle of work and energy to the block A after it collides with block B to find the velocity of the block A after its impact with B as follows:
Substitute
Substitute
Show the momentum impact diagram of the blocks A and B as in Figure (3).
The expression for the principle of conservation of momentum to the collision between the block A and block B as follows:
Here,
Substitute
Calculate the coefficient of restitution for the impact between the block A and block B
Substitute 0 for
Therefore, the coefficient of restitution between A and B
(b)
Find the displacement (x) of block C.
(b)
Answer to Problem 13.178P
The displacement (x) of block C is
Explanation of Solution
Given information:
The weight of the block A
The weight of the block B
The weight of the block C
The coefficient of friction between the block and plane
The initial speed of the block A
The blocks B and C are at rest.
The distance between the blocks (d) is
The width of the each blocks (b) is
The acceleration due to gravity (g) is
Calculation:
Show the diagram of the block B just before its impact with block C as in Figure (4).
The expression for the kinetic energy of the block B at position ‘2’ just after the impact with block A
The expression for the kinetic energy of the block B just before its impact with blocks C at the position ‘4’
Here,
The expression for the work done by the block B to overcome the frictional force in reaching position ‘4’ from position ‘2’ as follows:
The expression for the principle of work and energy to the block B just before its impact with block C at the position ‘2’ and position ‘4’ as follows:
Substitute
Substitute
Show the momentum impact diagram of the blocks B and C as in Figure (5).
The expression for the principle of conservation of momentum to the collision between the block B and block C as follows:
Substitute
Here,
Substitute
Calculate the coefficient of restitution for the impact between the block B and block C
Substitute 0 for
Show the diagram of the block C after its impact with Block B as in Figure (6).
The expression for the kinetic energy of the block C immediately after its impact with blocks B at position ‘4’
Finally, at the position ‘5’, the block C comes to rest. Thus,
The expression for the work done by the block C to overcome the frictional force in reaching the position ‘5’
Here, x is the distance travelled by the block C before coming to rest.
The expression for the principle of work and energy to the block C after its impact with block B as follows:
Substitute
Substitute
Therefore, displacement (x) of block C is
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Chapter 13 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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