Concept explainers
(a)
ToCalculate: The angular speed of the pulley.
(a)
Answer to Problem 86P
Explanation of Solution
Given information :
The circumference of the pulley
Mass of the pulley
Length of the rope
Mass of the rope
The difference in height of the two ends of the rope when the pulley system is at rest
Formula used :
Rotational kinetic energy
Where, Iis the moment of inertia,
Linear kinetic energy
Where, M is the mass and v is the speed.
Calculation:
Let the co-ordinate mechanism start at the center of the pulley, with the positive upward direction.
The conservation of energy to relate the final kinetic energy of the system
Initial kinetic energy is zero,
Final kinetic energy = K
Use the law of conservation of mechanical energy to find the angular velocity of the pulley when the difference in height between the two ends of the rope is 7.2 m.
By applying conservation of energy to relate the final kinetic energy of the system to the change in potential energy, can get,
The change in potential energy of the system is
Here,
Because
Substitute numerical values and evaluate
The kinetic energy of the system when the difference in height between the two ends of the rope is
Substitute values in equation:
Conclusion:
The angular speed of the pulley when the difference in height between the two ends of the rope is
(b)
ToFind: An expression for the
(b)
Answer to Problem 86P
Explanation of Solution
Given information :
The circumference of the pulley
Mass of the pulley
Length of the rope
Mass of the rope
The difference in height of the two ends of the rope when the pulley system is at rest
Formula Used :
Angular momentum:
Where, Iis the moment of inertia and
For the pulley
For the rope
Calculation:
The total angular momentum of the system is,
Letting
Assuming at
Substitute for
Solving for
Express
Integrate
Transform from logarithmic to exponential form to obtain:
Differentiate to express
Substitute for
Conclusion:
An expression for the angular momentum of the system as a function of time while neither end of the rope is above the center of the pulley is:
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