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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

The tension T in the string of the yo-yo in the figure is

T = m g R 2 r 2 + R 2

where m is the mass of the yo-yo and g is acceleration due to gravity. Use differentials to estimate the change in the tension if R is increased from 3 cm to 3.1 cm and r is increased from 0.7 cm to 0.8 cm. Does the tension increase or decrease?

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To determine

To estimate: The change in the tension T of the yo-yo by using differentials; infer whether the tension increase or decrease.

Explanation

Given:

The tension T in the string of the yo-yo is T=mgR2r2+R2 , where m is the mass of yo-yo, g is acceleration due to gravity, R is outer radius, r is inner radius.

The outer radius R=3cm .

The inner radius r=0.7cm .

The error in outer radius dR=0.1cm .

The error ininner radius dr=0.1cm .

Calculation:

Acoording to the given data outer radius and inner radius (R,r) change from (3,0.7) to (3.1,0.8) .

The change in the tension of yo-yo is calculated by, dT=TRdR+Trdr .

Take partial derivative of T with respect to R ,

TR=(2r2+R2)(mg)(1)mgR(2R)(2r2+R2)2[R(uv)=u'vv'uv2]=mg[2r2+R22R2](2r2+R2)2=mg(2r2R2)(2r2+R2)2

Take partial derivative of T with respect to r ,

TR=(2r2+R2)(0)mgR(2(2r))(2r2+R2)2

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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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