Ch 12: Rotation of a Rigid Body

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 12: Rotation of a Rigid Body

Problem 49b

Chapter 12, Problem 49b

A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. b. What is the rotational kinetic energy if the structure rotates about its center of mass at 100 rpm?

Verified step by step guidance

Determine the total mass of the system by adding the masses of the two balls and the rod. Convert all masses to kilograms for consistency.

Find the center of mass of the system. Use the formula for the center of mass:

x

_cm = (Σm_ix_i)/(Σm_i), where

m

_i is the mass of each component and

x

_i is its position relative to a reference point. Assume the rod is uniform and its mass is distributed evenly.

Calculate the moment of inertia of the system about the center of mass. Use the formula for the moment of inertia:

I = Σm

_ir_i², where

r

_i is the distance of each mass from the center of mass.

Convert the rotational speed from revolutions per minute (rpm) to radians per second using the conversion factor:

1 \, \(\text{rpm}\) = \(\frac{2π}{60}\) \, \(\text{rad/s}\)

Calculate the rotational kinetic energy using the formula:

KE

_rot = \(\frac{1}{2}\)Iω², where

I

is the moment of inertia and

ω

is the angular velocity in radians per second.

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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. This concept is crucial for understanding how the mass distribution and rotation speed of an object contribute to its overall energy.

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 of the object and the distribution of that mass relative to the axis of rotation. For composite systems, like the one in the question, the total moment of inertia is the sum of the individual moments of inertia of each component, calculated based on their distances from the axis of rotation.

Center of Mass

The center of mass is the point at which the mass of a system is balanced and can be considered to act. For a system of particles or rigid bodies, the center of mass can be found by taking the weighted average of their positions based on their masses. Understanding the center of mass is essential for calculating the moment of inertia and analyzing the rotational dynamics of the system.

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Related Practice

Textbook Question

A 75 g, 6.0-cm-diameter solid spherical top is spun at 1200 rpm on an axle that extends 1.0 cm past the edge of the sphere. The tip of the axle is placed on a support. What is the top's precession frequency in rpm?

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Textbook Question

A toy gyroscope has a ring of mass M and radius R attached to the axle by lightweight spokes. The end of the axle is distance R from the center of the ring. The gyroscope is spun at angular velocity ω, then the end of the axle is placed on a support that allows the gyroscope to precess. A 120 g, 8.0-cm-diameter gyroscope is spun at 1000 rpm and allowed to precess. What is the precession period?

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Textbook Question

Calculate the moment of inertia of the rectangular plate in FIGURE P12.55 for rotation about a perpendicular axis through the center.

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Textbook Question

Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when d=0 and when d = L/2.

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Textbook Question

Determine the moment of inertia about the axis of the object shown in FIGURE P12.52.

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Textbook Question

A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. a. How far is the center of mass from the 600 g ball?

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