Q. A particle of mass $m$ is fixed to one end of a light spring of force constant $k$ and unstretched length $l$. The system is rotated about the other end of the spring with an angular velocity $\omega$, in gravity free space. The increase in length of the spring will be:

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A

$\frac{m \omega^{2} l}{k}$

B

$\frac{m \omega^{2} l}{k-m \omega^{2}}$

C

$\frac{m \omega^{2} l}{k+m \omega^{2}}$

D

None of these

Solution:

Let increase in lengh of spring is $x$

At equilibrium,
$k x =m \omega^{2} x+m \omega^{2} l$
$k x-m \omega^{2} x =m \omega^{2} l $
$x =\frac{m \omega^{2} l}{k-m \omega^{2}}$

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Q. A particle of mass $m$ is fixed to one end of a light spring of force constant $k$ and unstretched length $l$. The system is rotated about the other end of the spring with an angular velocity $\omega$, in gravity free space. The increase in length of the spring will be:

$\frac{m \omega^{2} l}{k}$

$\frac{m \omega^{2} l}{k-m \omega^{2}}$

$\frac{m \omega^{2} l}{k+m \omega^{2}}$

None of these

Let increase in lengh of spring is $x$ At equilibrium, $k x =m \omega^{2} x+m \omega^{2} l$ $k x-m \omega^{2} x =m \omega^{2} l $ $x =\frac{m \omega^{2} l}{k-m \omega^{2}}$

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1. The work done by a particle moving with a velocity of $0.7\, c$ (where c is the velocity of light) in empty space free of electromagnetic field and far away from all matter is:

2. The machine gun fires $240 \, $ bullets per minute. If the mass of each bullet is $10 \, g$ and the velocity of the bullets is $600 \, m \, s^{- 1}$ ,the power (in $kW$ )of the gun is

4. A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_{c}$ is varying with time $t$ as $a_{c}=k^{2} \, rt^{2}$ , where $k$ is a constant. The power delivered to the particle by the force acting on it is

5. A ball of mass $1\, kg$ collides with a wall with speed $8\, ms ^{-1}$ and rebounds on the same line with the same speed. If mass of the wall is taken as infinite, the work done by the ball on the wall is

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1. If $E$ and $G$ respectively denote energy and gravitational constant, then $\frac{ E }{ G }$ has the dimensions of:

2. The de Broglie wavelength of an electron moving with kinetic energy of $144 \,eV$ is nearly

3. A car starts from rest and accelerates at $5\, m / s ^{2}$ At $t=4\, s$, a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at $t =6\, s$ ? (Take $\left. g =10\, m / s ^{2}\right)$

4. A wheel with $20$ metallic spokes each $1\, m$ long is rotated with a speed of $120\, rpm$ in a plane perpendicular to a magnetic field of $0.4\, G$. The induced emf between the axle and rim of the wheel will be, $(1 G = 10^{-4}\, T)$

5. A barometer is constructed using a liquid (density $= 760 \,kg/m^{3})$ What would be the height of the liquid column, when a mercury barometer reads $76\, cm$ ? (density of mercury $= 13600 kg/m^{3})$

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Let increase in lengh of spring is $x$

At equilibrium,
$k x =m \omega^{2} x+m \omega^{2} l$
$k x-m \omega^{2} x =m \omega^{2} l $
$x =\frac{m \omega^{2} l}{k-m \omega^{2}}$

5. A barometer is constructed using a liquid (density $= 760 \,kg/m^{3})$ What would be the height of the liquid column, when a mercury barometer reads $76\, cm$ ?
(density of mercury $= 13600 kg/m^{3})$