Laws of Motion & Friction · JEE Advanced previous year questions

A block of mass $5\text{ kg}$ moves along the $x$-direction subject to the force $F=(-20x+10)\text{ N}$, with the value of $x$ in metre. At time $t=0\text{ s}$, it is at rest at position $x=1\text{ m}$. The position and momentum of the block at $t=(\pi /4)\text{ s}$ are

1. A.$-0.5\text{ m},5\text{ kg m/s}$
2. B.$0.5\text{ m},0\text{ kg m/s}$
3. C.$0.5\text{ m},-5\text{ kg m/s}$
4. D.$-1\text{ m},5\text{ kg m/s}$

1. A.With a node at $\mathrm{O}$, the minimum frequency of vibration of the composite string is ${\nu }_0$.
2. B.With an antinode at $\mathrm{O}$, the minimum frequency of vibration of the composite string is $2{\nu }_0$.
3. C.When the composite string vibrates at the minimum frequency with a node at $\mathrm{O}$, it has $6$ nodes, including the end nodes.
4. D.No vibrational mode with an antinode at $\mathrm{O}$ is possible for the composite string.

A string of length $1\mathrm{m}$ and mass $2\times 10^{-5}\mathrm{kg}$ is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies $750\mathrm{Hz}$ and $1000\mathrm{Hz}$. The value of tension $T$ is ____ Newton.