Ch 10: Dynamics of Rotational Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 10: Dynamics of Rotational Motion

Problem 34e

Chapter 10, Problem 34e

An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airplane's engine is first started, it applies a constant torque of 1950 Nm to the propeller, which starts from rest. What is the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 revolutions?

Verified step by step guidance

First, convert the number of revolutions into radians. Since one revolution is \(2\pi\) radians, multiply 5.00 revolutions by \(2\pi\) to get the angular displacement in radians.

Next, use the work-energy principle. The work done by the torque is equal to the change in rotational kinetic energy. The work done \(W\) is given by \(W = \tau \theta\), where \(\tau\) is the torque and \(\theta\) is the angular displacement in radians.

Calculate the angular velocity \(\omega\) at the point when the propeller has turned through the given angular displacement. Use the equation \(\omega^2 = \omega_0^2 + 2\alpha\theta\), where \(\omega_0\) is the initial angular velocity (0 rad/s since it starts from rest), \(\alpha\) is the angular acceleration, and \(\theta\) is the angular displacement.

Determine the angular acceleration \(\alpha\) using the relation \(\tau = I\alpha\), where \(I\) is the moment of inertia of the propeller. For a rod rotating about its center, \(I = \frac{1}{12}mL^2\), where \(m\) is the mass and \(L\) is the length of the propeller.

Finally, calculate the instantaneous power output \(P\) of the motor using the formula \(P = \tau \omega\), where \(\tau\) is the torque and \(\omega\) is the angular velocity at the given instant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Torque

Torque is a measure of the rotational force applied to an object, causing it to rotate around an axis. It is calculated as the product of force and the distance from the axis of rotation, and is measured in Newton-meters (N•m). In this problem, the engine applies a constant torque to the propeller, initiating its rotation.

Angular Displacement

Angular displacement refers to the angle through which an object rotates around a fixed axis, measured in radians or revolutions. In this scenario, the propeller turns through 5.00 revolutions, which is crucial for calculating the angular velocity and subsequently the power output of the motor.

Power in Rotational Motion

Power in rotational motion is the rate at which work is done or energy is transferred in a rotating system. It is calculated as the product of torque and angular velocity. Understanding this concept is essential for determining the instantaneous power output of the motor when the propeller has completed 5.00 revolutions.

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Textbook Question

Compute the torque developed by an industrial motor whose output is 150 kW at an angular speed of 4000 rev/min.

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A 2.00-kg rock has a horizontal velocity of magnitude 12.0 m/s when it is at point P in Fig. E10.35. At this instant, what are the magnitude and direction of its angular momentum relative to point O?

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Textbook Question

A 2.00-kg rock has a horizontal velocity of magnitude 12.0 m/s when it is at point P in Fig. E10.35. If the only force acting on the rock is its weight, what is the rate of change (magnitude and direction) of its angular momentum at this instant?

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Textbook Question

A 2.80-kg grinding wheel is in the form of a solid cylinder of radius 0.100 m. What constant torque will bring it from rest to an angular speed of 1200 rev/min in 2.5 s?

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A woman with mass 50 kg is standing on the rim of a large disk that is rotating at 0.80 rev/s about an axis through its center. The disk has mass 110 kg and radius 4.0 m. Calculate the magnitude of the total angular momentum of the woman–disk system. (Assume that you can treat the woman as a point.)

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Textbook Question

An electric motor consumes 9.00 kJ of electrical energy in 1.00 min. If one-third of this energy goes into heat and other forms of internal energy of the motor, with the rest going to the motor output, how much torque will this engine develop if you run it at 2500 rpm?

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