Umakant Kondapure, Collin Fernandes, Nipun Bhatia, Vikram Bathula and, Ketki Deshpande Solutions for Chapter: Oscillations, Exercise 3: Competitive Thinking

A simple pendulum of length $\text{'}L\text{'}$ has mass $\text{'}M\text{'}$  and it oscillates freely with amplitude $\text{'}A\text{'}$. At extreme position, its potential energy is $(g=$ acceleration due to gravity)

The path length of oscillation of simple pendulum of length $1 \mathrm{meter}$ is$16 \mathrm{cm}$. Its maximum velocity is $\left(g={\mathrm{\pi }}^2 \mathrm{m} {\mathrm{s}}^{-2}\right)$,

A mass is suspended from a vertical spring which is executing SHM of frequency $5 \mathrm{Hz}$. The spring is unstretched at the highest point of oscillation. The maximum speed of the mass is (acceleration due to gravity, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$),

A simple pendulum attached to the ceiling of a stationary lift has a time period $T$ . The distance $y$ covered by the lift moving upwards varies with time $t$ as, $y=t^2$, where $y$ is in $\mathrm{metres}$ and $t$ in $\sec$. If $g=10 \mathrm{m} {\mathrm{s}}^{-2}$, then the time period of the pendulum will be,

When the kinetic energy of a body executing SHM is $\frac{1}{3}$ of the potential energy, the displacement of the body is $x$ percent of the amplitude where $x$ is,

Starting from the origin, a body oscillates simple harmonically with a period of $2 \mathrm{s}$. After what time will its kinetic energy be $75\%$ of the total energy?

A particle of mass $0.2 \mathrm{kg}$ moves in simple harmonic motion of amplitude $2 \mathrm{cm}$. If the total energy of the particle is $4\times {10}^{-5} \mathrm{J}$, then the time period of the motion is,

A point performs simple harmonic oscillation of period $T$ and the equation of motion is given by, $x=a\sin \left(\omega +\frac{\pi }{6}\right)$. After the elapsed of what fraction of the time period will the velocity of the point be equal to half its maximum velocity?