Concept explainers
(a)
The appropriate analysis model to describe the projectile and the rod
(a)
Answer to Problem 51AP
The appropriate analysis model to describe the particle and the rod is a projectile motion and the rod behaves like an isolated system.
Explanation of Solution
Conclusion:
The mass is moving to the right with linear velocity, as it strikes to a stationary rod the mass will follow projectile motion with the rod. The other end of the rod will also rotate about the fixed point to balance the rod, so the
Thus, the appropriate analytical model to describe the particle and the rod is a projectile motion and the rod behaves like an isolated system.
(b)
The angular momentum of the system before the collision about an axis through
(b)
Answer to Problem 51AP
The angular momentum of the system before the collision about an axis through
Explanation of Solution
The angular momentum before the collision for a given system is the sum of angular momentum of ball and rod. But before the collision angular momentum of the rod is zero because the rod is stationary.
Consider the mass of the ball is concentrated at its centre so it behaves like a particle.
Write the expression for the angular momentum of the system before the collision as.
Substitute
Re-arrange the terms.
Here,
Write the expression for momentum for a particle as.
Here,
Conclusion:
Substitute
Thus, the angular momentum of the system before the collision about an axis through
(c)
The moment of inertia of the system about an axis through
(c)
Answer to Problem 51AP
The moment of inertia of the system about an axis through
Explanation of Solution
The moment of inertia for a given system is the sum of the moment of inertia for particle and rod.
Write the expression for moment of inertia for a particle about the fix point
Here,
Write the expression for moment of inertia for rod about the fix point
Here,
Write the expression for total moment of inertia for rod-particle system about the fix point
Here,
Substitute
Simplify the above equation for
Conclusion:
Thus, the moment of inertia of the system about an axis through
(d)
The angular momentum of the system after the collision.
(d)
Answer to Problem 51AP
The total angular momentum of the system after the collision is
Explanation of Solution
Write the expression for the angular momentum of the system after the collision.
Here,
Conclusion:
Substitute
Thus, the total angular momentum of the system after the collision is
(e)
The angular speed
(e)
Answer to Problem 51AP
The angular speed
Explanation of Solution
Write the expression for conserved angular momentum as.
Here,
Conclusion:
Substitute
Simplify the above expression for
Thus, the angular speed
(f)
The kinetic energy of the system before the collision.
(f)
Answer to Problem 51AP
The kinetic energy of the particle before the collision is
Explanation of Solution
The kinetic energy of the system before the collision is equal to kinetic energy of the particle because before the collision the rod is not in motion so the kinetic energy of the rod becomes zero. Therefore before the collision the kinetic energy of the system becomes the kinetic energy of the particle.
Write the expression for kinetic energy of the particle before the collision as.
Here,
Conclusion:
Thus, the kinetic energy of the particle before the collision is
(g)
The kinetic energy of the system after the collision
(g)
Answer to Problem 51AP
The kinetic energy of the system after the collision is
Explanation of Solution
Write the expression for rotational kinetic energy of the particle-rod system after the collision as.
Here,
Conclusion:
Substitute
Simplify the above expression as.
Thus, the kinetic energy of the system after the collision is
(h)
The fractional change of kinetic energy due to the collision.
(h)
Answer to Problem 51AP
The fractional change of kinetic energy due to the collision is
Explanation of Solution
Write the expression for change in kinetic energy for the system as.
Here,
Write the expression for fractional change in kinetic energy as.
Here,
Conclusion:
Substitute
Simplify the above expression for
Substitute
Thus, the fractional change of kinetic energy due to the collision is
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Chapter 11 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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