When the rope is at an angle of α = 30°, the 1-lb sphere A has a speed v0 = 4 ft/s. The coefficient of restitution between A and the 2-lb wedge B is 0.7 and the length of rope l = 2.6 ft. The spring constant has a value of 2 lb/in. and θ = 20°. Determine (a) the velocities of A and B immediately after the impact, (b) the maximum deflection of the spring, assuming A does not strike B again before this point.
(a)
Find the velocity of A
Answer to Problem 13.188P
The velocity of A
Explanation of Solution
Given information:
The angle of the rope
The weight of the sphere A
The weight of the wedge B
The speed of the sphere A
The coefficient of restitution between A and wedge (e) is 0.7.
The length of the rope (l) is
The spring constant (k) is
The angle
The acceleration due to gravity (g) is
Calculation:
Calculate the mass of sphere A
Substitute
Calculate the mass of wedge B
Substitute
Calculate the initial altitude of sphere
Substitute
Calculate the initial potential energy of sphere
Substitute
Calculate the initial kinetic energy of sphere
Here,
Substitute
Calculate the altitude of sphere just before impact
Substitute
Calculate the initial potential energy of sphere just before impact
Substitute
Calculate the kinetic energy of sphere just before impact
Here,
Substitute
The expression for the principle of conservation of energy between initial and final stage of sphere as follows:
Substitute
Show the impulse-momentum diagram for sphere as in Figure (1).
The expression for the momentum in tangential direction as follows:
Here,
Substitute
Show the impulse-momentum diagram of wedge as in Figure (2).
The expression for the momentum in x-direction as follows:
Here,
Substitute
Calculate the coefficient of restitution (e) using the formula:
Substitute 0.7 for e,
Find the velocity of sphere B immediately after the impact:
Substitute
Find the normal component of velocity of sphere:
Substitute
Calculate the resultant velocity of sphere A
Substitute
Calculate the angle for
Substitute
Calculate the resultant angle of velocity of sphere A
Substitute 63.77 for
Therefore, the velocity of A
(b)
Find the maximum deflection of the spring assuming A does not strike B again before this point.
Answer to Problem 13.188P
The maximum deflection of the spring
Explanation of Solution
Given information:
The angle of the rope
The weight of the sphere A
The weight of the wedge B
The speed of the sphere A
The coefficient of restitution between A and wedge (e) is 0.7.
The length of the rope (l) is
The spring constant (k) is
The angle
The acceleration due to gravity (g) is
Calculation:
Calculate the kinetic energy of wedge block just before impact
Here,
Substitute
The expression for the potential energy of spring at the end of impact
The expression for the principle of conservation of energy for wedge block as follows:
Here,
Substitute
Substitute
Therefore, the maximum deflection of the spring
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Chapter 13 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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