Springs

Q. A load suspended by a massless spring produces an extension of $x\text{cm}$ in equilibrium. When it is cut into two unequal parts, the same load produces an extension of $7.5\text{cm}$ when suspended by the larger part of length $60\text{cm}$. When it is suspended by the smaller part, the extension is $5.0\text{cm}$. Then

1. $x=12.5\text{cm}$
2. $x=3.0\text{cm}$
3. The original length of the spring is $90\text{cm}$.
4. The original length of the spring is $80\text{cm}$

Q. Two wires of the same material (Young's modulus $Y$) and same length $L$ but radii $R$ and $2R$ respectively are joined end to end and a weight $w$ is suspended from the combination as shown in the figure. The elastic potential energy stored in the system is

1. $\frac{3w^{2}L}{4\pi R^{2}Y}$
2. $\frac{3w^{2}L}{8\pi R^{2}Y}$
3. $\frac{5w^{2}L}{8\pi R^{2}Y}$
4. $\frac{w^{2}L}{\pi R^{2}Y}$

Q. Two wires of the same material (Young's modulus $Y$) and same length $L$ but radii $R$ and $2R$ respectively are joined end to end and a weight $w$ is suspended from the combination as shown in the figure. The elastic potential energy stored in the system is

1. $\frac{3w^{2}L}{4\pi R^{2}Y}$
2. $\frac{3w^{2}L}{8\pi R^{2}Y}$
3. $\frac{5w^{2}L}{8\pi R^{2}Y}$
4. $\frac{w^{2}L}{\pi R^{2}Y}$

Q. Two springs $P$ and $Q$ of force constants $k_P$ and $k_Q(k_Q=\frac{k_P}{2})$ are stretched by applying forces of equal magnitude. If the energy stored in $Q$ is $E$, then the energy stored in $P$ is

1. $E$
2. $\frac{E}{8}$
3. $\frac{E}{4}$
4. $\frac{E}{2}$