University Physics with Modern Physics (14th Edition)
University Physics with Modern Physics (14th Edition)
14th Edition
ISBN: 9780321973610
Author: Hugh D. Young, Roger A. Freedman
Publisher: PEARSON
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Textbook Question
Chapter 9, Problem 9.1DQ

Which of the following formulas is valid if the angular acceleration of an object is not constant? Explain your reasoning in each ease, (a) υ = (b) atan = (c) ω = ω0 + αt; (d) atan = 2; (e) K = 1 2 I ω 2 .

(a)

Expert Solution
Check Mark
To determine

The relation v=rω is valid or not if the angular acceleration of an object is not constant.

Explanation of Solution

The relation for displacement is,

s=rθ                                              (I)

s is displacement.

r is the radius of circular path

θ is angular distance.

Relation v=rω is derived from the equation (I).

The relation s=rθ doesn’t depend on whether angular acceleration is constant or not. Thus, if an object doesn’t have a constant acceleration it will not affect its velocity. Hence relation v=rω is valid.

Conclusion:

The relation v=rω is valid.

(b)

Expert Solution
Check Mark
To determine

The relation atan=rα  is valid or not if the angular acceleration of an object is not constant.

Explanation of Solution

The expression for tangential acceleration in terms of angular acceleration is,

atan=rα

atan is tangential acceleration.

α is angular acceleration.

Tangential acceleration is possessed by the object when it moves along the curve. The angular acceleration also doesn’t affect it. Thus relation atan=rα is valid.

Conclusion:

The relation atan=rα is valid.

(c)

Expert Solution
Check Mark
To determine

The relation ω=ω0+αt is valid or not if the angular acceleration of an object is not constant.

Explanation of Solution

The expression for angular velocity is,

ω=ω0+αt .

ω0 is initial angular velocity.

t is the time.

ω is the final angular velocity.

The above expression is derived from the assumption that the angular acceleration is constant. Thus, relation ω=ω0+αt is not valid.

Conclusion:

The relation ω=ω0+αt is not valid.

(d)

Expert Solution
Check Mark
To determine

The relation atan=rω2 is valid or not if the angular acceleration of an object is not constant.

Explanation of Solution

The expression for tangential acceleration in terms of angular velocity is,

atan=rω2

For an object that moves in a circular path then it has centripetal acceleration and it doesn’t depends on the whether angular acceleration is constant or not. Thus above relation is valid. Hence the relation atan=rω2 is valid.

Conclusion:

The relation atan=rω2 is valid.

(e)

Expert Solution
Check Mark
To determine

The relation K=12Iω2 is valid or not if the angular acceleration of an object is not constant.

Explanation of Solution

The expression for kinetic energy is,

K=12Iω2                                    (II)

K is kinetic energy.

I is moment of inertia.

The equation (II) is derived from,

K=12mv2

m is mass.

Substitute rω for v in above expression to find K .

K=12m(rω)2=12mr2ω2=12Iω2

The relation rω is valid for any acceleration. Thus K=12Iω2 is valid.

Conclusion:

The relation K=12Iω2 is valid.

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Chapter 9 Solutions

University Physics with Modern Physics (14th Edition)

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