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Chapter 15, Problem 1CQ
To determine

(a) To Determine:

What is the period (T2) for a block oscillating on a spring when the block’s mass (m) is doubled?

Expert Solution
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Answer to Problem 1CQ

Solution:

The period (T2) for a block oscillating on a spring when the block’s mass (m) is doubled is 2.8 s

Explanation of Solution

Given:

The period T1 of an oscillating block is 2 s.The new mass, m2 is double the previous mass m1, or m2=2m1.

Formula used:

The period T of a block of mass m oscillating on a spring with constant k is defined as:

T=2πmk                                        (1)

Where:

m is the mass of block.

k is spring constant.

T is the period.

Calculation:

The period (T1) for mass (m1) according to equation (1) is:

T1=2πm1k                                       (2)

Now, when the mass is doubled, period (T2) is:

T2=2π2m1k=2(2πm1k)           (3)

Inserting (2) into equation (3) we have:

T2=2 (T1)                                    (4)

But:

T1=2 s                                               (5)

Inserting (5) into (4) we obtain:

T2=2.8 s                                             (6)

To determine

(b) To Determine:

What is the period (T2) for a block oscillating on a spring when the constant spring (k) is quadrupled?

Expert Solution
Check Mark

Answer to Problem 1CQ

Solution:

When the constant spring (k) is quadrupled, period (T2) is 1 s.

Explanation of Solution

Given:

The period (T1) of an oscillating block is 2 s. The spring constant (k) is quadrupled.

According to equation (1), period (T1)  is:

T1=2πmk1                                   (7)

Now, when the spring constant (k1) is quadrupled, period (T2) is:

T2=2πm4k1=12 (2πmk1)                      (8)

Replacing (7) into equation (8) we have:

T2=12 T1                                               (9)

But T1=2 s, inserting this value into equation (9) we obtain:

T2=1                                          (10)

To determine

(c) To determine:

What is the period (T2) if the oscillation amplitude (A) is doubled while (m) and (k) are unchanged?

Expert Solution
Check Mark

Answer to Problem 1CQ

Solution:

Explanation of Solution

Given:

The oscillation amplitude (A) is doubledwhile (m) and (k) are unchanged.

According to equation (1) the period does not depend on oscillation amplitude (A). If the mass (m) and (k) are unchanged, then the period does not change either.

Conclusion:

From the equations (6) and (10) it can be established that if the mass of a block tied to a spring is doubled (for a period T1 of 2 s) then period (T2) is 2.8 s. If the spring constant is quadrupled (for a period T1 of 2 s) then period (T2) is 1s. Similarly if the oscillation amplitude is doubled then the period T2 does not change.

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Chapter 15 Solutions

Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (Chs 1-42) Plus Mastering Physics with Pearson eText -- Access Card Package (4th Edition)

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