A boy ties a stone of mass 100 g to the end of a 2 m long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of 80 N . If the maximum speed with which the stone can revolve is $\frac { K } { \pi } \mathrm { rev } \min ^ { - 1 }$. The value of $K$ is : (Assume the string is massless and un-stretchable)
(1) 400
(2) 300
(3) 600
(4) 800

A particle is moving with constant speed in a circular path. When the particle turns by an angle $90^{\circ}$, the ratio of instantaneous velocity to its average velocity is $\pi : x\sqrt{2}$. The value of $x$ will be
(1) 2
(2) 5
(3) 1
(4) 7

A coin placed on a rotating table just slips when it is placed at a distance of 1 cm from the centre. If the angular velocity of the table is halved, it will just slip when placed at a distance of $\_\_\_\_$ from the centre:
(1) 8 cm
(2) 4 cm
(3) 1 cm
(4) 2 cm

A vehicle of mass 200 kg is moving along a levelled curved road of radius 70 m with angular velocity of $0.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centripetal force acting on the vehicle is:
(1) 560 N
(2) 2800 N
(3) 2240 N
(4) 14 N

A small block of mass 100 g is tied to a spring of spring constant $7.5\mathrm{~N~m}^{-1}$ and length 20 cm. The other end of spring is fixed at a particular point $A$. If the block moves in a circular path on a smooth horizontal surface with constant angular velocity $5\mathrm{~rad~s}^{-1}$ about point $A$, then tension in the spring is
(1) 0.75 N
(2) 0.25 N
(3) 0.50 N
(4) 1.5 N

A bob of mass $m$ is suspended by a light string of length $L$. It is imparted a minimum horizontal velocity at the lowest point $A$ such that it just completes half circle reaching the top most position $B$. The ratio of kinetic energies $\frac { ( \text { K.E. } ) _ { A } } { ( \text { K.E. } ) _ { B } }$ is:
(1) $3 : 2$
(2) $5 : 1$
(3) $2 : 5$
(4) $1 : 5$

A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution?
(1) 25
(2) 50
(3) 100
(4) 20

A ball of mass 0.5 kg is attached to a string of length 50 cm. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is 400 N. The maximum possible value of angular velocity of the ball in rad $\mathrm { s } ^ { - 1 }$ is,:
(1) 1600
(2) 40
(3) 1000
(4) 20

A particle is moving in a circle of radius 50 cm in such a way that at any instant the normal and tangential components of its acceleration are equal. If its speed at $\mathrm { t } = 0$ is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the time taken to complete the first revolution will be $\frac { 1 } { \alpha } \left[ 1 - \mathrm { e } ^ { - 2 \pi } \right] \mathrm { s }$, where $\alpha =$ $\_\_\_\_$ .

Q3. A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius 9 m and completes 120 resolutions in 3 minutes. The magnitude of centripetal acceleration of monkey is (in $\mathrm { m } / \mathrm { s } ^ { 2 }$ ):
(1) $57600 \pi ^ { 2 } \mathrm {~ms} ^ { - 2 }$
(2) Zero
(3) $4 \pi ^ { 2 } \mathrm {~ms} ^ { - 2 }$
(4) $16 \pi ^ { 2 } \mathrm {~ms} ^ { - 2 }$

Q6. A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is : ( Given $=$ Radius of geo-stationary orbit for earth is $4.2 \times 10 ^ { 4 } \mathrm {~km}$ )
(1) $1.4 \times 10 ^ { 4 } \mathrm {~km}$
(2) $1.05 \times 10 ^ { 4 } \mathrm {~km}$
(3) $8.4 \times 10 ^ { 4 } \mathrm {~km}$
(4) $1.68 \times 10 ^ { 5 } \mathrm {~km}$

Q7. Two planets $A$ and $B$ having masses $m _ { 1 }$ and $m _ { 2 }$ move around the sun in circular orbits of $r _ { 1 }$ and $r _ { 2 }$ radii respectively. If angular momentum of $A$ is $L$ and that of $B$ is 3 L , the ratio of time period $\left( \frac { T _ { A } } { T _ { B } } \right)$ is:
(1) $\left( \frac { r _ { 2 } } { r _ { 1 } } \right) ^ { \frac { 3 } { 2 } }$
(2) $\frac { 1 } { 27 } \left( \frac { m _ { 2 } } { m _ { 1 } } \right) ^ { 3 }$
(3) $27 \left( \frac { m _ { 1 } } { m _ { 2 } } \right) ^ { 3 }$
(4) $\left( \frac { r _ { 1 } } { r _ { 2 } } \right) ^ { 3 }$

A body of mass $m$ connected to a massless and unstretchable string goes in a vertical circle of radius $R$ under gravity $g$. The other end of the string is fixed at the center of circle. If velocity at top of circular path is $n\sqrt{gR}$, where $n \geq 1$, then ratio of kinetic energy of the body at bottom to that at top of the circle is:
(1) $\frac{n^2}{n^2 + 4}$
(2) $\frac{n^2 + 4}{n^2}$
(3) $\frac{n+4}{n}$
(4) $\frac{n}{n+4}$

Two disc having same moment of inertia about their axis. Thickness is $t _ { 1 }$ and $t _ { 2 }$ and they have same density. If $R _ { 1 } / R _ { 2 } = 1 / 2$, then find $t _ { 1 } / t _ { 2 }$.
(A) 4
(B) $1 / 4$
(C) $1 / 16$
(D) 16

A block of mass m is at rest w.r.t. hollow cylinder which is rotating with angular speed $\omega$. Radius of cylinder is R. Find minimum coefficient of friction between block and cylinder.
(A) $\frac{g}{4\omega^2 R}$
(B) $\frac{3g}{2\omega^2 R}$
(C) $\frac{g}{\omega^2 R}$
(D) $\frac{2g}{\omega^2 R}$