Two identical steel balls, each of diameter 25.4 nun and moving in opposite directions at 5 m/s, run into each other head-on and bounce apart. Prior to the collision, one of the balls is squeezed in a vise while precise measurements are made of the resulting amount of compression. The results show that Hooke’s law is a fair model of the ball’s elastic behavior. For one datum, a force of 16 kN exerted by each jaw of the vise results in a 0.2-mm reduction in the diameter. The diameter returns to its original value when the force is removed, (a) Modeling the ball as a spring, find its spring constant. (b) Does the interaction of the balls during the collision last only for an instant or for a nonzero time interval? State your evidence, (c) Compute an estimate for the kinetic energy of each of the balls before they collide, (d) Compute an estimate for the maximum amount of compression each ball undergoes when the balls collide, (e) Compute an order-of-magnitude estimate for the time interval for which the balls are in contact. (In Chapter 15, you will learn to calculate the contact time interval precisely.)
(a)
The spring constant of ball when models the ball as a spring.
Answer to Problem 7.57AP
The spring constant of ball is
Explanation of Solution
Given info: The diameter of each steel ball is
The ball models as a spring then from Hooke’s law, the force exerts on the spring is,
Here,
Rearrange the above equation.
Substitute
Conclusion:
Therefore, the spring constant of ball is
(b)
The interaction of the balls during the collision.
Answer to Problem 7.57AP
The interaction of the balls during the collision cannot happen.
Explanation of Solution
The interaction of the balls during the collision is calculable through the time period.
From the Newton’s law, the force exerts by a jaw is,
Here,
The expression for the acceleration of the steel balls is,
Here,
The total velocity of the balls during the collision is,
Here,
Substitute
Substitute
The interaction of the balls during the collision lasts for a time interval if the interaction takes no time interval that means
Substitute
The zero time interval of interaction, the force exerted by each ball on the other would be infinite and that cannot happen.
Conclusion:
Therefore, the interaction of the balls during the collision cannot happen.
(c)
The kinetic energy of each of the balls before they collide.
Answer to Problem 7.57AP
The kinetic energy of each ball is
Explanation of Solution
The value of density of the steel is
The expression for the volume of steel ball is,
Here,
The radius of ball is the half of its diameter so the radius of steel ball is,
Here,
Substitute
Substitute
The expression for the mass of the steel ball is,
Substitute
The expression for the kinetic energy of the steel ball before they collide is,
Substitute
Conclusion:
Therefore, the kinetic energy of each ball is
(d)
The maximum amount of compression each balls when balls collide.
Answer to Problem 7.57AP
The maximum amount of compression of each balls is
Explanation of Solution
From part (c), the kinetic energy before collision of balls is,
The expression for the kinetic energy of spring after the collision of balls is,
Here,
The ball is models as a spring so the kinetic energy before the collisions of balls is converted into spring energy.
From part (a), the spring constant of balls that models as a spring is,
Substitute
Conclusion:
Therefore, the maximum amount of compression of each balls is
(e)
The time interval for which the balls are in contact.
Answer to Problem 7.57AP
The time interval is
Explanation of Solution
From part (a), the force exerts on the both the balls that models as spring is,
The average force of both the balls is,
From part (b), the expression for the time interval is,
Substitute
The steel balls move in opposite direction of each other so the velocity has opposite sign after they bounce apart to each other.
From part (a), the spring constant of balls that models as a spring is,
From part (c), the mass of steel ball is,
From part (d), the spring constant of the ball is,
Substitute
Conclusion:
Therefore, the time interval is
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Chapter 7 Solutions
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