(a)
The angular speed of the disk once pure rolling takes place.
(a)
Answer to Problem 61CP
The angular speed of the disk once pure rolling takes place is
Explanation of Solution
For the particle under the net force model the net force will be the rate of change of linear momentum of the body.
Write the expression for net force on the disc .
Here,
The initial velocity of the disc is zero.
Substitute
For the particle under the net torque model the net torque will be the rate of change of angular momentum of the body.
Write the expression for net torque on the disc as.
Simplify the above expression for
Here,
Substitute
Simplify the above-obtained expression for
Write the expression for final linear velocity of the disc.
Substitute
Simplify equation (V) for
Re-arrange the terms.
Simplify the above expression for
Write the expression for moment of inertia of the disc as.
Conclusion:
Substitute
Simplify the above expression for
Thus, the angular speed of the disk once pure rolling takes place is
(b)
The fractional change in kinetic energy from the moment the disk is set down until pure rolling occurs.
(b)
Answer to Problem 61CP
The fractional change in kinetic energy from the moment the disk is set down until pure rolling occurs
Explanation of Solution
Write the expression for the rotational kinetic energy of the disc as.
Here,
Write the expression for kinetic energy due to the linear velocity of the disc as
Here,
The change in kinetic energy is the difference between initial and final kinetic energy.
Initially, the disc is only rotating so initial kinetic energy is rotational kinetic energy. In the final condition, the disc has linear and angular velocities so the kinetic energy becomes the sum of linear and rotational kinetic energy.
Write the expression for the change in kinetic energy of the disc as.
Here,
Divide equation (X) by equation (IX) it gives.
Conclusion:
Substitute
Simplify the above expression for
Thus, the fractional change in kinetic energy from the moment the disk is set down until pure rolling occurs
(c)
The time interval after setting the disk down before pure rolling motion begins.
(c)
Answer to Problem 61CP
The time interval after setting the disk down before pure rolling motion begins is
Explanation of Solution
Write the expression for time interval after setting the disk down before pure rolling motion begins as.
Here,
Initially, the linear velocity of the disc is zero therefore the initial linear momentum of the disc becomes zero. As the initial momentum is zero, the change in linear momentum equal to the final linear momentum of the disc.
Write the expression for change in linear momentum as.
Write the expression for friction force between disc and plane surface as.
Here,
Substitute
Conclusion:
Substitute
Simplify the above obtained expression for
Thus, the time interval after setting the disk down before pure rolling motion begins is
(d)
The distance travel by the disc before pure rolling begins.
(d)
Answer to Problem 61CP
The distance travel by the disc before pure rolling begins is
Explanation of Solution
Write the expression for average velocity of the disc as.
Initial linear velocity of the disc is zero.
Substitute
Substitute
Here,
Write the expression for the disc under constant acceleration as.
Here,
Conclusion:
Substitute
Substitute
Simplify the above expression for
Thus, the distance travel by the disc before pure rolling begins 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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