Question

Consider a spring that exerts the following restoring force: $F=-kx$ for $x>0$ and $F=-4kx$ for $x<0$
A mass m on a frictionless surface is attached to the spring, displaced to $x=A$ by stretching the spring and then releasing:

• A. The period of motion will be $T=\frac{3}{2}\pi \sqrt{\frac{m}{k}}$
• B. The most negative value, the mass $m$ can reach will be $x=-\frac{A}{2}$
• C. The time taken to move from $x=A$ to $x=-\frac{A}{\sqrt{2}}$, straight away will be equal to $\frac{5\pi }{8}\sqrt{\frac{m}{k}}$
• D. The total energy of oscillations will be $\frac{5}{2}k{A}^2$

Answer 1

Answer: (A), (B)

The period of motion will be $T=\frac{3}{2}\pi \sqrt{\frac{m}{k}}$
The most negative value, the mass $m$ can reach will be $x=-\frac{A}{2}$

We know that the formula of kinetic energy and time period in oscillation
hence, kinetic energy = time period of oscillation in both the cases
$T=\frac{1}{2}(2\pi \sqrt{\frac{m}{k}}+2\pi \sqrt{\frac{m}{4k}})$
$T=\frac{3}{2}\pi \sqrt{\frac{m}{k}}$
$\frac{1}{2}K{A}^2=\frac{1}{2}4K{A}^{\prime 2}$
$A^{\prime }=A/2$
$A^{\prime }$ is the new amplitude