Question

An ideal spring is connected to the center of a solid cylinder. Initially the spring is compressed and compression is $40cm$. Now the cylinder is released from rest. Find the speed of the top point of the cylinder when the spring gains its relaxed length. (assuming friction between cylinder and ground is sufficient for pure rolling or rolling without slipping)

• A. $10.4m/s$
• B. $10.4m/s$
• C. $18.2m/s$
• D. $22.4m/s$

Answer 1

The correct option is A $10.4m/s$

Apply conservation of energy,
$\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2=\frac{1}{2}K{x}^2$
$\frac{1}{2}m{v}^2+\frac{1}{2}\frac{m{r}^2}{2}.\frac{v^2}{r^2}=\frac{1}{2}K{x}^2$
$\frac{3}{4}m{v}^2=\frac{1}{2}K{x}^2$
$v^2=\frac{4}{6}\frac{K{x}^2}{m}=\frac{2K{x}^2}{3m}$
$v=x\sqrt{\frac{2K}{3m}}=0.4\sqrt{\frac{2\times 1000}{3\times 4}}$
$v=0.4\sqrt{\frac{500}{3}}$
$=0.4\sqrt{170}\simeq 5.2m/s$
Speed of top point$=2v=10.4m/s$