Concept explainers
(a)
The point moves a greater distance in a given time, if the disk rotates with increasing
(a)
Answer to Problem 1P
The particle on the rim covers a greater linear distance as compared to the particle situated half way between the rim and the axis of rotation.
Explanation of Solution
Given:
A disk rotating with increasing angular velocity about an axis passing through its center and perpendicular to its plane.
Distance of the particle 1 (on the rim) from the axis of rotation
Distance of particle 2 (half way between the axis of rotation and the rim) from the axis of rotation
Here, the radius of the disk is
Formula used:
The
Here,
The angular acceleration of a point on the rotating disk is related to the linear acceleration
Here,
The linear distance
Calculations:
The disk of radius
from the axis of rotation. This is shown in the figure 1 below:
Figure 1
If the disk rotates with an angular acceleration
Calculate the linear accelerations of the particles located at point 1 and 2 using the equation (2).
Substitute
Therefore, from the above equations,
If the disk is assumed to start from rest, both particles would start with their initial velocities
Use equation (3) to calculate the distance travelled by the two points.
From equation (5),
Conclusion:
Therefore, the point on the rim travels a greater distance when compared to the point located halfway between the rim and the axis of rotation.
(b)
The point that turns through a greater angle.
(b)
Answer to Problem 1P
Both the points turn through the same angle.
Explanation of Solution
Formula used:
The
Here
Calculation:
The points 1 and 2 located at points A and B on the disc rotating with an angular acceleration
Figure 2
At any instant of time, both particles have the same instantaneous angular velocity and angular acceleration. As it can be seen from Figure 2, both particles describe the same angle at an instant of time.
Conclusion:
Thus, the particle located at the rim and the particle located half way between the rim and the axis of rotation turn through the same angle.
(c)
The point which travels with greater speed.
(c)
Answer to Problem 1P
The point on the rim travels with greater speed.
Explanation of Solution
Formula used:
The instantaneous speed
Here,
Calculation:
The disk moves with increasing angular velocity. But, both points at any instant would have the same instantaneous angular velocity, since they turn through the same angle in a given interval of time.
Hence, it can be inferred from equation (7):
Since the point on the rim has the greater value of
Conclusion:
Thus, the point on the rim would have a greater speed when compared to the point located half way between the rim and the axis of rotation.
(d)
The point which has the greater angular speed.
(d)
Answer to Problem 1P
Both the particles have the same angular speed.
Explanation of Solution
Formula used:
The angular velocity of a particle is given by
Calculation:
From Figure 2, it is seen that at any instant of time, both particles 1 and 2 cover the same angles. Hence, the rate of change of their angular displacement
Conclusion:
Thus, the particle on the rim and the particle located halfway between the rim and the axis of rotation have the same angular velocity.
(e)
The point which has the greater tangential acceleration.
(e)
Answer to Problem 1P
The point on the rim has a greater tangential acceleration when compared to the point located midway between the rim and the axis of rotation:
Explanation of Solution
Formula used:
The tangential acceleration
Calculation:
If the disk rotates with a varying angular velocity, it has angular acceleration. Assuming that the angular acceleration of the disk remains constant, from equation (9) it can be inferred that
The point on the rim has the greater value of
Therefore, the point on the rim has a greater tangential acceleration when compared to any point located inside the rim. This is also proved by the fact that the point on the rim gains a larger tangential velocity when compared to any inner point.
Conclusion:
Thus, the point on the rim has a greater tangential acceleration when compared to the point located midway between the rim and the axis of rotation.
(f)
The point which has a greater angular acceleration.
(f)
Answer to Problem 1P
Both particles have the same angular acceleration.
Explanation of Solution
Formula used:
The angular acceleration
Calculation:
It has been proved in (d) that at any instant of time, both the particles have the same angular velocity. Therefore, in an interval of time
Hence, from equation (1), it can be proved that at a given instant of time, both the points will have the same angular accelerations.
Conclusion:
Thus, both particles are found to have the same angular acceleration.
(g)
The point which has the greater centripetal acceleration.
(g)
Answer to Problem 1P
The point on the rim has a greater centripetal acceleration.
Explanation of Solution
Formula used:
The centripetal acceleration of a point located at a distance
Calculation:
It has been established in part (d) that at any instant of time, the point on the rim and the point located halfway between the rim and the axis of rotation have the same angular velocity.
Therefore, from equation (10), it can be inferred that
The point on the rim has the greater value of
Conclusion:
Thus, the point on the rim has a greater centripetal acceleration.
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Chapter 9 Solutions
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