Principles of Physics: A Calculus-Based Text
Principles of Physics: A Calculus-Based Text
5th Edition
ISBN: 9781133104261
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 12, Problem 60P

(a)

To determine

Prove that the maximum stretching of spring from mean position is x=μsmgk.

(a)

Expert Solution
Check Mark

Answer to Problem 60P

It is proved that the maximum stretching of spring from mean position is x=μsmgk.

Explanation of Solution

Write the relation between force on spring and maximum static frictional force.

    kx=μsn

Here, k is the spring force, x is the displacement of spring, μs is the coefficient of static friction, and n is the normal reaction force.

Write the expression for n.

  n=mg

Here, m is the mass of block and g is the gravitational acceleration.

Rewrite the first equation by substituting the above one.

    kx=μsmg

Conclusion:

Rewrite the expression for x.

    x=μsmgk

Therefore, it is proved that the maximum stretching of spring from mean position is x=μsmgk.

x=μsmgk.

(b)

To determine

Prove that block oscillates about mean point when spring is stretched by x=μsmgk.

(b)

Expert Solution
Check Mark

Answer to Problem 60P

It is proved that block oscillates about mean point when spring is stretched by x=μsmgk.

Explanation of Solution

The block will be almost at rest at the break point because of the smaller value of velocity. The block starts to oscillate s soon when the force on block becomes kx and +μkmg. Write the expression for net force on block while sliding.

    Fnet=kx+μkmg

Here, Fnet is the net force on block,.

Rewrite the above equation by multiplying and dividing the second term with k.

    Fnet=kx+kμkmgk=k(xμkmgk)

Express the term μkmgk into a single variable.

    xμkmgk=xrel

Here, xrel is the displacement of block from the point μkmgk.

Rewrite the expression for Fnet by substituting the above relation.

    Fnet=k(xrel)

Conclusion:

From the above calculation, it is evident that block starts to be in simple harmonic motion about mean position on displacing the spring

Therefore, it is proved that block oscillates about mean point when spring is stretched by x=μsmgk.

(c)

To determine

Plot the variation of position of block with time.

(c)

Expert Solution
Check Mark

Answer to Problem 60P

The plot is

Principles of Physics: A Calculus-Based Text, Chapter 12, Problem 60P , additional homework tip  1

Explanation of Solution

The graph is plotted with time on x-axis and position on y-axis.

Principles of Physics: A Calculus-Based Text, Chapter 12, Problem 60P , additional homework tip  2

The dotted line along x-axis denotes the function xrel=μkmgk. The peak on curve denotes x=μsmgk

Conclusion:

Therefore, the graph is plotted

(d)

To determine

Prove that the amplitude of oscillation is A=(μsμk)mgk.

(d)

Expert Solution
Check Mark

Answer to Problem 60P

It is proved that the amplitude of oscillation is A=(μsμk)mgk.

Explanation of Solution

Write the expression for xrel from part (b).

    xrel=xμkmgk

Rewrite the above expression by substituting μsmgk for x.

    xrel=μsmgkμkmgk=(μsμk)mgk

Amplitude is the maximum displacement from the mean position only. Rewrite the expression as amplitude.

    A=(μsμk)mgk

Here, A is the amplitude.

On pulling the block to distance x=μsmgk, it will start to oscillate about x=μkmgk.

Conclusion:

Therefore, it is proved that the amplitude of oscillation is A=(μsμk)mgk.

(e)

To determine

Prove that the period of oscillation is T=2(μsμk)mgvk+πmk.

(e)

Expert Solution
Check Mark

Answer to Problem 60P

It is proved that the period of oscillation is T=2(μsμk)mgvk+πmk.

Explanation of Solution

Write the expression for time taken by block to move with the board

    T1=2Av

Here, T1 is the time period and v is the speed of motion.

Rewrite the expression by substituting (μsμk)mgk for A.

    T1=2((μsμk)mgk)v

Time taken by block to move from mean position to an extreme position.

    T1/2=12(2πmk)=πmk

Here, T1/2 is the time taken to complete the half of SHM (same as that of time needed to spring back the block to original position.

The period of oscillation is the sum of T1 and T1/2.

    T=T1+T1/2

Here, T is the time period of oscillation.

Conclusion:

Rewrite the above equation by substituting the relations for T1 and T1/2.

    T=2((μsμk)mgk)v+πmk=2((μsμk)mg)kv+πmk

Therefore, It is proved that the period of oscillation is T=2(μsμk)mgvk+πmk.

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Chapter 12 Solutions

Principles of Physics: A Calculus-Based Text

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