Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Question

Chapter 5.5, Problem 52E

To determine

To calculate the value of dT .

Expert Solution

Answer to Problem 52E

dT=−π(Lg3)12dg

Explanation of Solution

Given information:

The relation of T , g , L is given by T=2π(Lg)12

Calculation:

Obtain derive of T in terms of g .

T=2π(Lg)12dTdg=2πL12(−12g−32)dT=−πL12g−32dgdT=−π(Lg3)12dg

To determine

To explain that if g increases, will T increase or decrease, will a pendulum clock speed up or slow down.

Expert Solution

Answer to Problem 52E

If g increases then T decreases and as T decreases the pendulum will speed up.

Explanation of Solution

Given information:

The relation of T , g , L is given by T=2π(Lg)12

According to part (a) you have dT=−π(Lg3)12dg , where (Lg3)12 is always positive because it is a root. Hence, the value of dT will be negative if dg is positive and will be positive if dg is negative. If g increased, dg is positive. Hence dT is negative, therefore T is decreased.

Since the time period T is the time it takes to complete a swing. The more is the period the less is the speed. As T decreased pendulum will speed up.

To determine

To find dg and estimate the value of g at the new location.

Expert Solution

Answer to Problem 52E

The value of dg is −0.977 cm/sec2 and the value of g is 979.023 cm/sec2

Explanation of Solution

Given information:

A clock with 100 cm pendulum is moved from a location where g=980 cm/sec2 to a new location. This increases the time period by dT=0.001 sec .

Calculation:

According to part (a) you have dT=−π(Lg3)12dg ,

Now substitute the values in the above equation

dT=−π(Lg3)12dg0.001=−π(1009803)12dgdg=−0.977

For g ,

g=980+dgg=980−0.977g=979.023

Therefore,

The value of dg is −0.977 cm/sec2 and the value of g is 979.023 cm/sec2

Chapter 5.5, Problem 51E

Chapter 5.5, Problem 53E

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

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