Question

A mass m attached to a spring of spring constant k is stretched by a distance $x_0$ from its equilibrium position and released with no initial velocity, on a smooth horizontal surface. The maximum speed attained by mass in its subsequent motion and the time at which this speed would be attained are respectively:

• A. $\sqrt{\frac{k}{m}}{x}_0$ and $\pi \sqrt{\frac{m}{k}}$
• B. $\sqrt{\frac{k}{m}}\frac{x_0}{2}$ and $\frac{\pi }{2}\sqrt{\frac{m}{k}}$
• C. $\sqrt{\frac{k}{m}}{x}_0$ and $\frac{\pi }{2}\sqrt{\frac{m}{k}}$
• D. $\sqrt{\frac{k}{m}}\frac{x_0}{2}$ and $\pi \sqrt{\frac{m}{k}}$

Answer 1

Answer: (C)

$\sqrt{\frac{k}{m}}{x}_0$ and $\frac{\pi }{2}\sqrt{\frac{m}{k}}$

At the mean position, the speed will be maximum

$\frac{k{x}_0^2}{2}=\frac{m{v}^2}{2}\Rightarrow v_{max}=\sqrt{\frac{k}{m}}{x}_0$

and this is attained at $t=\frac{T}{4}$.

Time period of motion is $T=2\pi \sqrt{\frac{m}{k}}$

So required time is $t=\frac{\pi }{2}\sqrt{\frac{m}{k}}$