Concept explainers
(a)
The speed of each point in meters per second.
(a)
Answer to Problem 32PQ
The speed of the point at the outer end of the blade having length
Explanation of Solution
Write the equation connecting linear speed and the angular speed of a point.
Here,
It is given that both the blades have the same angular speed.
Use equation (I) to find the expression for the linear speed of the point at the outer end of the blade having length
Here,
Use equation (I) to find the expression for the linear speed of the point at the outer end of the blade having length
Here,
Conclusion:
It is given that the angular speed is
Substitute
Substitute
Therefore, the speed of the point at the outer end of the blade having length
(b)
The angular distance travelled by each point.
(b)
Answer to Problem 32PQ
The angular distance travelled by each point is
Explanation of Solution
Write the equation for the
Here,
Rewrite the above equation for
Equation (IV) shows that angular distance travelled by each point is same since both the blades have same angular speed.
Conclusion:
It is given that the motion is considered for
Substitute
Therefore, the angular distance travelled by each point is
(c)
The translational distance travelled by each point.
(c)
Answer to Problem 32PQ
The translational distance travelled by the point at the outer end of the blade having length
Explanation of Solution
The translational distance travelled by each point will be equal to the arc length subtended by the angular displacement of each point.
Write the equation for the arc length.
Here,
Use equation (V) to find the expression for translational distance travelled by the point at the outer end of the blade having length
Here,
Use equation (V) to find the expression for translational distance travelled by the point at the outer end of the blade having length
Here,
Conclusion:
Substitute
Substitute
Therefore, the translational distance travelled by the point at the outer end of the blade having length
(d)
The magnitude of centripetal acceleration that would be experienced by an object located at each point.
(d)
Answer to Problem 32PQ
The magnitude of centripetal acceleration that would be experienced by an object located at the point at the outer end of the blade having length
Explanation of Solution
Write the equation for the centripetal acceleration.
Here,
Use equation (VIII) to find the expression for the centripetal acceleration that would be experienced by an object located at the point at the outer end of the blade having length
Here,
Use equation (VIII) to find the expression for the centripetal acceleration that would be experienced by an object located at the point at the outer end of the blade having length
Here,
Conclusion:
Substitute
Substitute
Therefore, the magnitude of centripetal acceleration that would be experienced by an object located at the point at the outer end of the blade having length
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