Concept explainers
A uniform disk, initially at rest and of constant thickness, is placed in contact with the belt shown, which moves at a constant speed v = 80 ft/s. Knowing that the coefficient of kinetic friction between the disk and the belt is 0.15, determine (a) the number of revolutions executed by the disk before it reaches a constant angular velocity, (b) the time required for the disk to reach that constant angular velocity.
Fig. P17.135
(a)
Find the number of revolutions executed by the disk before it reaches a constant angular velocity.
Answer to Problem 17.135RP
The number of revolutions executed by the disk is
Explanation of Solution
Given information:
The speed of the belt is
The coefficient of kinetic friction between the disk and the belt is
The radius of the uniform disk is
Calculation:
Show the free-body diagram of the disk as in Figure 1.
Find the frictional force
Here, the normal force is N.
Substitute 0.15 for
Resolve the vertical component of forces to find the normal force.
Here, the mass of the disk is m and the acceleration due to gravity is g.
Consider the acceleration due to gravity as
Substitute 0.15N for
Substitute
Find the mass moment of inertia
Here, the radius of the disk is r.
Find the angular velocity
Find the work done
Here, the number of revolutions is
Substitute
The total kinetic energy at initial position is zero.
Find the total kinetic energy after reaching the constant angular velocity using the relation.
Substitute
Write the equation of conservation of energy and work as follows;
Substitute 0 for
Substitute 5 in. for r,
Therefore, the number of revolutions executed by the disk is
(b)
Find the time required for the disk to reach the constant angular velocity.
Answer to Problem 17.135RP
The time required for the disk is
Explanation of Solution
Given information:
The speed of the belt is
The coefficient of kinetic friction between the disk and the belt is
The radius of the uniform disk is
Calculation:
Show the free-body diagram of the principle of impulse-momentum as in Figure 2.
Take moment about point A as follows;
Substitute
Substitute
Therefore, the time required for the disk is
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