Ch 09: Rotation of Rigid Bodies

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 09: Rotation of Rigid Bodies

Problem 38

Chapter 9, Problem 38

An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm (rev/min) about an axis through its center. You can model the propeller as a slender rod.
(a) What is its rotational kinetic energy?
(b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to 75.0% of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

Verified step by step guidance

Step 1: Understand the problem and identify the given values. The propeller is modeled as a slender rod, which means its moment of inertia can be calculated using the formula for a slender rod rotating about its center: I = (1/12) * m * L^2, where m is the mass and L is the length. The given values are: length L = 2.08 m, mass m = 117 kg, and angular velocity ω = 2400 rpm. Convert ω to radians per second using the relation ω (rad/s) = ω (rpm) * (2π / 60).

Step 2: Calculate the moment of inertia (I) of the propeller using the formula I = (1/12) * m * L^2. Substitute the values of m and L into the formula to find I.

Step 3: Use the formula for rotational kinetic energy, K = (1/2) * I * ω^2, to calculate the rotational kinetic energy of the propeller. Substitute the values of I and ω (converted to rad/s) into the formula to find K.

Step 4: For part (b), reduce the mass to 75.0% of its original value. The new mass m' = 0.75 * m. To maintain the same rotational kinetic energy, use the formula K = (1/2) * I' * ω'^2, where I' = (1/12) * m' * L^2. Solve for the new angular velocity ω' in rad/s using the equation K = (1/2) * I' * ω'^2.

Step 5: Convert the new angular velocity ω' from rad/s back to rpm using the relation ω' (rpm) = ω' (rad/s) * (60 / 2π). This gives the angular speed in rpm required to maintain the same kinetic energy with the reduced mass.

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Key Concepts

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

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by an object due to its rotation. It is calculated using the formula KE_rot = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity in radians per second. For a slender rod rotating about its center, the moment of inertia is I = (1/12) m L², where m is the mass and L is the length of the rod. Understanding this concept is crucial for calculating the kinetic energy of the airplane propeller.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a slender rod rotating about its center, the moment of inertia is given by I = (1/12) m L². This concept is essential for determining how the mass and shape of the propeller affect its rotational kinetic energy and angular speed.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second or revolutions per minute (rpm). It is related to the rotational kinetic energy and moment of inertia, as changes in mass or shape can affect the required angular velocity to maintain a specific kinetic energy. In this problem, understanding how to convert between mass and angular velocity while keeping kinetic energy constant is key to solving part (b).

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Related Practice

Textbook Question

You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is 60.0 cm long and has mass 0.400 kg. What is the moment of inertia of this rod for an axis at its center, perpendicular to the rod?

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Textbook Question

A uniform sphere with mass $28.0$ kg and radius $0.380$ m is rotating at constant angular velocity about a stationary axis that lies along a diameter of the sphere. If the kinetic energy of the sphere is $236$ J, what is the tangential velocity of a point on the rim of the sphere?

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Textbook Question

A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at t = 0, the wheel turns through 8.20 revolutions in 12.0 s. At t = 12.0 s the kinetic energy of the wheel is 36.0 J. For an axis through its center, what is the moment of inertia of the wheel?

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Textbook Question

You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is 60.0 cm long and has mass 0.400 kg. One of your engineers has proposed to reduce the moment of inertia by bending the rod at its center into a V-shape, with a 60.0^o angle at its vertex. What would be the moment of inertia of this bent rod about an axis perpendicular to the plane of the V at its vertex?

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Textbook Question

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass M and radius R about an axis perpendicular to the hoop's plane at an edge.

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Textbook Question

A wagon wheel is constructed as shown in Fig. E9.33. The radius of the wheel is 0.300 m, and the rim has mass 1.40 kg. Each of the eight spokes that lie along a diameter and are 0.300 m long has mass 0.280 kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel? (Use Table 9.2.)

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