Question

You hang a mass Mfrom a spring, which stretches an amount ${\mathit{S}}_{\mathbf{1}}$. Then you cut the spring in half, and hang a mass Mfrom one half. How much does the half-spring stretch? (a) $\mathbf{2}{\mathit{s}}_{\mathbf{1}}$, (b) ${\mathit{S}}_{\mathbf{1}}$,(c)${\mathit{s}}_{\mathbf{1}}\mathbf{/}\mathbf{2}$.

Short answer

The half-spring stretches by$S_1/2$.

Step-by-step solution

Stating given data

A mass M is hung from a string and the initial displacement is $S_1$.

Defining spring constant and displacement

The spring constant k represents the stiffness of the spring. Stiffer (more difficult to stretch) springs have higher spring constants. The displacement of an object is a measurement of the distance that describes the change from a normal or equilibrium position.

The spring constant k of a spring doubles if it is cut in half.

The displacement of a spring when a force F is applied to it is defined using the following formula:

$x=\frac{F}{k}$

Determining change in spring displacement

Since the spring constant doubles if the length is halved, the new spring constant is 2k. if the initial spring constant was k, 2k.

From equation (I), the change in displacement of the spring is

$$\begin{gathered} x=\frac{F}{2k} \\ =\frac{1}{2}\frac{F}{k} \end{gathered}$$

Substitute the displacement $S_1$for $\frac{F}{k}$in the above equation.

$$\begin{gathered} x=\frac{1}{2}{s}_1 \\ =\frac{s_1}{2} \end{gathered}$$

Hence, the new displacement is $\frac{s_1}{2}$.