Concept explainers
A 1.8-kg collar A and a 0.7-kg collar B can slide without friction on a frame, consisting of the horizontal rod OE and the vertical rod CD, which is free to rotate about its vertical axis of symmetry. The two collars are connected by a cord running over a pulley that is attached to the frame at O. At the instant shown, the velocity vA of collar A has a magnitude of 2.1 m/s and a stop prevents collar B from moving. The stop is suddenly removed and collar A moves toward E. As it reaches a distance of 0.12 m from O, the magnitude of its velocity is observed to be 2.5 m/s. Determine at that instant the magnitude of the angular velocity of the frame and the moment of inertia of the frame and pulley system about CD.
Fig. P17.89
Find the magnitude of the angular velocity of the frame and the moment of inertia of the frame and pulley system about CD.
Answer to Problem 17.89P
The magnitude of the angular velocity of the frame is
The magnitude of the moment of inertia of the frame and pulley system about CD is
Explanation of Solution
Given information:
The mass
The mass
The velocity
The distance
The velocity
Calculation:
Write the equation of the velocity component
Here,
Write the equation of the velocity
Here,
Let
Let
Write the equation of the velocity component
Substitute
Find the angular velocity
Substitute
The velocity of the collar A in radial direction at initial position
Find the equation of the kinetic energy
Here, I is the moment of inertia of the frame,
Substitute
At initial position, potential energy will be zero. Therefore,
Write the equation of the angular momentum
Substitute
Let
Write the equation of the velocity
Here,
Substitute 0.12 m for
Find the equation of the velocity component
Substitute
Find the velocity
Find the change in radial direction or constraint in collar A
Substitute 0.12 m for
Find the constraint in collar B using Equation (2).
Substitute
Find the equation of the kinetic energy
Substitute 1.8 kg for
Find the potential energy
Substitute 0.7 kg for
Find the angular momentum at final position
Substitute 1.8 kg for
Consider the conservation of angular momentum.
Substitute
Take
Consider the conservation of energy.
Substitute
Substitute
Substitute
Solve equation (4),
Thus, the magnitude of the moment of inertia of the frame and pulley system about CD
Find the magnitude of the angular velocity of the frame
Substitute
Thus, the magnitude of the angular velocity of the frame is
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