Concept explainers
(a)
The rotational kinetic energy.
(a)
Answer to Problem 54P
The rotational kinetic energy for the system is
Explanation of Solution
Redraw the figure P10.54.
Consider that the vertically standing to be initial position and horizontal to be the final position.
Write the equation for conservation of energy.
Here,
From the law of conservation of energy, gain in rotational kinetic energy equals to loss in gravitational potential energy for the given system.
Write the expression for rotational kinetic energy.
Here,
Write the expression for loss in gravitational potential energy for sphere.
Here,
Write the expression for loss in gravitational potential energy for rod.
Here,
Substitute
Conclusion:
Substitute
Thus, the rotational kinetic energy for the system is
(b)
The angular speed of the rod and ball.
(b)
Answer to Problem 54P
The angular speed of the ball and the rod is
Explanation of Solution
Write the expression for moment of inertia of sphere at center.
Here,
Write the expression for the parallel axis theorem for moment of inertia at point
Here,
Substitute
Write the expression for moment of inertia of rod at point
Here,
Write the expression for net moment of inertia for the whole system.
Here,
Substitute
Write the expression for rotational kinetic energy.
Here,
Simply the above equation for value of
Conclusion:
Substitute
Substitute
Thus, the angular speed of the ball and the rod is
(c)
Thelinear speed of the center of mass of the ball.
(c)
Answer to Problem 54P
The linear speed of the ball of center of mass is
Explanation of Solution
Write the expression for linear speed of the ball.
Here,
Substitute
Here,
Conclusion:
Substitute
Thus, the linear speed of the ball of center of mass is
(d)
Compare the speed with the speed had the ball fallen freelythrough the same distance of
(d)
Answer to Problem 54P
The rod pulls the sphere down together while rotating by the speed factor
Explanation of Solution
Loss in gravitational potential energy will be equal to gain in kinetic energy.
Write the expression for the conservation of energy.
Write the expression for loss in gravitational potential energy for sphere.
Here,
Write the expression for gain kinetic energy.
Here,
Substitute
Write the expression for the ratio of new speed to the original speed.
Here,
Conclusion:
Substitute
Substitute
Thus, the rod pulls the sphere down together while rotating by more speed than in direct falling by the factor of
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