Mechanics Questions

A uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. The extension $x_0$ of the spring when it is in equilibrium is:

Assume that a drop of liquid evaporates by decrease in its surface energy, so that its temperature remains unchanged. What should be the minimum radius of the drop for this to be possible? The surface tension is $T$, density of liquid is $\rho$ and $L$ is its latent heat of vaporization.

Let $[{ϵ}_0]$ denote the dimensional formula of the permittivity of vacuum. If $M=\text{mass},L=\text{length},T=\text{time}$ and $A=\text{electric current}$, then:

A sonometer wire of length $1.5\text{m}$ is made of steel. The tension in it produces an elastic strain of $1\%$. What is the fundamental frequency of steel if density and elasticity of steel are $7.7\times 10^3{\text{kg/m}}^3$ and $2.2\times 10^{11}{\text{N/m}}^2$ respectively?

What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in a circular orbit at an altitude of $2R$?

A projectile is given an initial velocity of $(\hat{i}+2\hat{j})\text{m/s}$, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g=10{\text{m/s}}^2$, the equation of its trajectory is:

A hoop of radius $r$ and mass $m$ rotating with an angular velocity ${\omega }_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

Statement – I : A point particle of mass $m$ moving with speed $v$ collides with stationary point particle of mass $M$. If the maximum energy loss possible is given as $f(\frac{1}{2}m{v}^2)$, then $f=\frac{m}{M+m}$.Statement – II : Maximum energy loss occurs when the particles get stuck together as a result of the collision.

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5\text{s}$. In another $10\text{s}$ it will decrease to $\alpha$ times its original magnitude, where $\alpha$ equals: