Concept explainers
An elevator system in a tall building consists of a 800-kg car and a 950-kg counterweight joined by a light cable of constant length that passers over a pulley of mass 280 kg. The pulley, called a sheave, is a solid cylinder of radius 0.700 m turning on a horizontal axle. The cable does not slip on the sheave. A number n of people, each of mass 80.0 kg, are riding in the elevator car, moving upward at 3.00 m/s and approaching the floor where the car should stop. As an energy-conservation measure, a computer disconnects the elevator motor at just the right moment so that t he sheave–car–counterweight system then coasts freely without friction and comes to rest at the floor desired. There it is caught by a simple latch rather than by a massive brake. (a) Determine the distance d the car coasts upward as a function of n. Evaluate the distance for (b) n = 2, (c) n = 12, and (d) n = 0. (e) For what integer values of n does the expression in part (a) apply? (f) Explain your answer to part (e). (g) If an infinite number of people could fit on the elevator, what is the value of d?
(a)
The distance
Answer to Problem 10.68AP
The distance
Explanation of Solution
The mass of car is
From the law of energy conservation,
Here,
Formula to calculate the total initial energy of the system is,
Here,
Write the expression for the initial translational kinetic energy of the elevator is,
Here,
Write the expression for the initial translational kinetic energy of the counterweight is,
Here,
Write the expression for the initial rotational kinetic energy of the sheave is,
Here,
Write the expression for the moment of inertia of the pulley is,
Here,
Write the expression for the initial angular speed of the pulley is,
Here,
Substitute
Substitute
Since at the end the system comes to rest hence all the kinetic energies will be zero only potential energy remains in the system.
Formula to calculate the total final energy of the system is,
Here,
Write the expression for the final potential energy of the elevator is,
Here,
Write the expression for the final potential energy of the counterweight is,
Here,
Since the sheave pulley remains at its position so its final potential energy is zero.
Substitute
Substitute
Formula to calculate the mass of the elevator is,
Here,
Substitute
Substitute
Conclusion:
Therefore, the distance
(b)
The distance
Answer to Problem 10.68AP
The distance
Explanation of Solution
The expression for the distance
Substitute
Conclusion:
Therefore, the distance
(c)
The distance
Answer to Problem 10.68AP
The distance
Explanation of Solution
The expression for the distance
Substitute
Conclusion:
Therefore, the distance
(d)
The distance
Answer to Problem 10.68AP
The distance
Explanation of Solution
The expression for the distance
Substitute
Conclusion:
Therefore, the distance
(e)
The integral values of
Answer to Problem 10.68AP
The expression in part (a) is valid only when
Explanation of Solution
The expression for the distance
From the above expression, the distance
Conclusion:
Therefore, the expression in part (a) is valid only when
(f)
The explanation for the answer in part (e).
Answer to Problem 10.68AP
The mass of the elevator is less than the mass of the counterweight for the value of
Explanation of Solution
The expression for the distance
Substitute
Since the value of distance
Conclusion:
Therefore, the mass of the elevator is less than the mass of the counterweight for the value of
(g)
The value of
Answer to Problem 10.68AP
The value of
Explanation of Solution
The expression for the distance
Rearrange the above equation.
Substitute
Since the value of distance
Conclusion:
Therefore, the value of
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