Question: User: 6
8 The top ends of two springs, S₁ and S₂, are attached to a rod.
rod
S₁
S₂
x
x/3
A mass is hung from the bottom end of S₁. The extension of S₁ is $x$. The elastic potential energy in the spring is $E$. The same mass is hung from the bottom end of S₂. The extension of S₂ is $\frac{x}{3}$.
What is the elastic potential energy in the spring S₂?
A $\frac{E}{9}$
B $\frac{E}{3}$
C $3E$
D $9E$
Your answer
[1]
top-center: rod at the top center; S₁ on the left below the rod with a downward extension labeled $x$; S₂ on the right below the rod with a downward extension labeled $\frac{x}{3}$; both springs are hanging vertically from the rod with masses at the bottom.
1. **State the problem:**
We have two springs, $S_1$ and $S_2$, attached to a rod. A mass hung on $S_1$ causes an extension $x$ and elastic potential energy $E$. The same mass hung on $S_2$ causes an extension $\frac{x}{3}$. We want to find the elastic potential energy in $S_2$.
2. **Recall the formula for elastic potential energy:**
$$E = \frac{1}{2} k x^2$$
where $k$ is the spring constant and $x$ is the extension.
3. **Important rule:**
The elastic potential energy depends on the square of the extension.
4. **Given:**
For $S_1$:
$$E = \frac{1}{2} k_1 x^2$$
For $S_2$:
Extension is $\frac{x}{3}$, so energy is:
$$E_2 = \frac{1}{2} k_2 \left(\frac{x}{3}\right)^2 = \frac{1}{2} k_2 \frac{x^2}{9} = \frac{k_2}{k_1} \cdot \frac{E}{9}$$
5. **Find relation between $k_1$ and $k_2$:**
Since the same mass causes extension $x$ in $S_1$ and $\frac{x}{3}$ in $S_2$, the forces are equal:
$$mg = k_1 x = k_2 \frac{x}{3}$$
Divide both sides by $x$:
$$k_1 = \frac{k_2}{3}$$
6. **Substitute $k_1 = \frac{k_2}{3}$ into energy expression:**
$$E = \frac{1}{2} k_1 x^2 = \frac{1}{2} \frac{k_2}{3} x^2 = \frac{k_2 x^2}{6}$$
Rearranged:
$$k_2 x^2 = 6E$$
7. **Calculate $E_2$:**
$$E_2 = \frac{1}{2} k_2 \left(\frac{x}{3}\right)^2 = \frac{1}{2} k_2 \frac{x^2}{9} = \frac{k_2 x^2}{18}$$
Substitute $k_2 x^2 = 6E$:
$$E_2 = \frac{6E}{18} = \frac{E}{3}$$
**Final answer:**
The elastic potential energy in spring $S_2$ is $\frac{E}{3}$.
**Answer choice:** B