A small 4-lb collar C can slide freely on a thin ring of weight 6 lb and radius 10 in. The ring is welded to a short vertical shaft, which can rotate freely in a fixed bearing. Initially, the ring has an angular velocity of 35 rad/s and the collar is at the top of the ring (θ = 0) when it is given a slight nudge. Neglecting the effect of friction, determine (a) the angular velocity of the ring as the collar passes through the position θ = 90°, (b) the corresponding velocity of the collar relative to the ring.
Fig. P17.91
(a)
Find the angular velocity of the ring as the collar passes through the position
Answer to Problem 17.91P
The angular velocity of the ring as the collar passes through the position
Explanation of Solution
Given information:
The weight
The weight
The initial angular velocity
Calculation:
Find the mass
Substitute 4 lb for
Find the mass
Substitute 6 lb for
Sketch the component of velocity of the given system at initial position as shown in Figure (1).
Find the centroidal moment of inertia
Here, R is the radius of the ring.
The collar C is located at top of the ring at initial position. Therefore, the angle
Sketch the component of velocity of the given system at final position as shown in Figure (2).
Refer Figure (2).
Write the equation of the velocity
Here,
Consider the conservation of angular momentum.
Substitute
Simplify the Equation.
Substitute
Thus, the angular velocity of the ring as the collar passes through the position
(b)
Find the corresponding velocity of the collar relative to the ring.
Answer to Problem 17.91P
The corresponding velocity of the collar relative to the ring is
Explanation of Solution
Calculation:
Find the equation of the kinetic energy
Substitute
Find the equation of the potential energy
Find the equation of the kinetic energy
Substitute
The potential energy of the system at final position
Consider the conservation of energy.
Substitute
Substitute
Thus, the corresponding velocity of the collar relative to the ring is
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