Chapter 15, Problem 1CQ

To determine

(a) To Determine:

What is the period $\left(T_2\right)$ for a block oscillating on a spring when the block’s mass $\left(m\right)$ is doubled?

Answer to Problem 1CQ

Solution:

The period $\left(T_2\right)$ for a block oscillating on a spring when the block’s mass $\left(m\right)$ is doubled is 2.8 s

Explanation of Solution

Given:

The period $T_1$ of an oscillating block is 2 s.The new mass, $m_2$ is double the previous mass$m_1$, or$m_2=2m_1$.

Formula used:

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

Where:

• $m$ is the mass of block.

• $k$ is spring constant.

• $T$ is the period.

Calculation:

The period $\left(T_1\right)$ for mass $\left(m_1\right)$ according to equation (1) is:

Now, when the mass is doubled, period $\left(T_2\right)$ is:

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

But:

Inserting (5) into (4) we obtain:

To determine

(b) To Determine:

What is the period $\left(T_2\right)$ for a block oscillating on a spring when the constant spring $\left(k\right)$ is quadrupled?

Answer to Problem 1CQ

Solution:

When the constant spring $\left(k\right)$ is quadrupled, period $\left(T_2\right)$ is 1 s.

Explanation of Solution

Given:

The period $\left(T_1\right)$ of an oscillating block is 2 s. The spring constant ($k$) is quadrupled.

According to equation (1), period $\left(T_1\right)$ is:

Now, when the spring constant $\left(k_1\right)$ is quadrupled, period $\left(T_2\right)$ is:

$T_2=2\pi \sqrt{\frac{m}{4k_1}}=\frac{1}{2} \left(2\pi \sqrt{\frac{m}{k_1}}\right) \left(8\right)$

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

But$T_1=2 \text{s}$, inserting this value into equation (9) we obtain:

To determine

(c) To determine:

What is the period $\left(T_2\right)$ if the oscillation amplitude $\left(A\right)$ is doubled while $\left(m\right)$ and $\left(k\right)$ are unchanged?

Answer to Problem 1CQ

Solution:

Explanation of Solution

Given:

The oscillation amplitude $\left(A\right)$ is doubledwhile $\left(m\right)$ and $\left(k\right)$ are unchanged.

According to equation (1) the period does not depend on oscillation amplitude$\left(A\right)$. If the mass $\left(m\right)$ and $\left(k\right)$ 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 $T_1$ of 2 s) then period $\left(T_2\right)$ is 2.8 s. If the spring constant is quadrupled (for a period $T_1$ of 2 s) then period $\left(T_2\right)$ is 1s. Similarly if the oscillation amplitude is doubled then the period $T_2$ does not change.