Ch 10: Dynamics of Rotational Motion

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 10: Dynamics of Rotational Motion

Problem 41a

Chapter 10, Problem 41a

A hollow, thin-walled sphere of mass $12.0 kg$ and diameter $48.0 cm$ is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by $θ (t) = A t^{2} + B t^{4}$ , where A has numerical value $1.50$ and B has numerical value $1.10$ . What are the units of the constants A and B?

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To determine the units of the constants A and B, we need to analyze the equation θ(t) = At^2 + Bt^4. The angle θ is measured in radians, which are dimensionless. Therefore, the units of At^2 and Bt^4 must also be dimensionless.

Consider the term At^2. Since t is time measured in seconds (s), t^2 has units of s^2. For At^2 to be dimensionless, the units of A must be the inverse of s^2, which is s^-2.

Now, consider the term Bt^4. Similarly, t^4 has units of s^4. For Bt^4 to be dimensionless, the units of B must be the inverse of s^4, which is s^-4.

Thus, the units of A are s^-2, and the units of B are s^-4.

This analysis ensures that each term in the equation θ(t) = At^2 + Bt^4 is dimensionally consistent, resulting in a dimensionless angle θ.

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Key Concepts

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

Angular Displacement

Angular displacement is the angle through which a point or line has been rotated in a specified sense about a specified axis. It is measured in radians, which is a dimensionless unit. In the given function θ(t) = At^2 + Bt^4, θ represents the angular displacement as a function of time.

Dimensional Analysis

Dimensional analysis involves checking the consistency of units in equations. It helps determine the units of constants by ensuring that both sides of an equation have the same dimensions. For θ(t) = At^2 + Bt^4, the units of A and B can be found by ensuring the terms At^2 and Bt^4 have units of radians.

Rotational Motion

Rotational motion refers to the motion of a body around a center or axis. It involves concepts like angular velocity and angular acceleration. In this problem, understanding how angular displacement changes with time is crucial, as it relates to the rotational dynamics of the sphere described by θ(t) = At^2 + Bt^4.

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Related Practice

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. At this instant, what are the magnitude and direction of its angular momentum relative to point O?

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Textbook Question

A small block on a frictionless, horizontal surface has a mass of 0.0250 kg. It is attached to a massless cord passing through a hole in the surface (Fig. E10.40). The block is originally revolving at a distance of 0.300 m from the hole with an angular speed of 2.85 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.150 m. Model the block as a particle. Find the change in kinetic energy of the block.

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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 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.)