The minimum velocity (in $\mathrm{ms}^{-1}$) with which a car driver must traverse a flat curve of radius 150 m and coefficient of friction 0.6 to avoid skidding is
(1) 60
(2) 30
(3) 15
(4) 25

Consider a car moving on a straight road with a speed of $100 \mathrm{~m}/\mathrm{s}$. The distance at which car can be stopped is $[\mu_\mathrm{k} = 0.5]$
(1) 800 m
(2) 1000 m
(3) 100 m
(4) 400 m

A particle is moving in a circular path of radius $a$, with a constant velocity $v$ as shown in the figure. The centre of circle is marked by 'C'. The angular momentum from the origin O can be written as:
(1) $va(1+\cos 2\theta)$
(2) $va(1+\cos\theta)$
(3) $va\cos 2\theta$
(4) $va$

A particle is moving along a circular path with a constant speed of $10 \mathrm {~ms} ^ { - 1 }$. What is the magnitude of the change in velocity of the particle, when it moves through an angle of $60 ^ { \circ }$ around the centre of the circle?
(1) $10 \sqrt { 3 } \mathrm {~m} / \mathrm { s }$
(2) zero
(3) $10 \sqrt { 2 } \mathrm {~m} / \mathrm { s }$
(4) $10 \mathrm {~m} / \mathrm { s }$

A smooth wire of length $2 \pi r$ is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed $\omega$ about the vertical diameter AB, as shown in figure, the bead is at rest with respect to the circular ring at position P as shown. Then the value of $\omega ^ { 2 }$ is equal to:
(1) $2 \mathrm {~g} / \mathrm { r }$
(2) $\frac { \sqrt { 3 } \mathrm {~g} } { 2 \mathrm { r } }$
(3) $2 g / ( r \sqrt { 3 } )$
(4) $( \mathrm { g } \sqrt { 3 } ) / \mathrm { r }$

A body of mass $\mathrm { m } = 10 \mathrm {~kg}$ is attached to one end of a wire of length 0.3 m . What is the maximum angular speed (in $\mathrm { rad } \mathrm { s } ^ { - 1 }$ ) with which it can be rotated about its other end in a space station without breaking the wire? [Breaking stress of wire $( \sigma ) = 4.8 \times 10 ^ { 7 } \mathrm {~N} \mathrm {~m} ^ { - 2 }$ and area of cross-section of the wire $= 10 ^ { - 2 } \mathrm {~cm} ^ { 2 }$ ]

A thin circular ring of mass $M$ and radius $r$ is rotating about its axis with an angular speed $\omega$. Two particles having mass $m$ each are now attached at diametrically opposite points. The angular speed of the ring will become:
(1) $\omega \frac { M } { M + m }$
(2) $\omega \frac { M + 2 m } { M }$
(3) $\omega \frac { M } { M + 2 m }$
(4) $\omega \frac { M - 2 m } { M + 2 m }$

A particle of mass $m$ is suspended from a ceiling through a string of length $L$. The particle moves in a horizontal circle of radius $r$ such that $r = \frac{L}{\sqrt{2}}$. The speed of particle will be:
(1) $\sqrt{rg}$
(2) $\sqrt{2rg}$
(3) $\sqrt{\frac{rg}{2}}$
(4) $2\sqrt{rg}$

A mass $M$ hangs on a massless rod of length $l$ which rotates at a constant angular frequency. The mass $M$ moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular momentum of $M$ about point $A$ is $L _ { A }$ which lies in the positive $z$ direction and the angular momentum of $M$ about $B$ is $L _ { B }$. The correct statement for this system is:
(1) $L _ { A }$ and $L _ { B }$ are both constant in magnitude and direction
(2) $L _ { B }$ is constant in direction with varying magnitude
(3) $L _ { B }$ is constant, both in magnitude and direction
(4) $L _ { A }$ is constant, both in magnitude and direction

Two identical particles of mass 1 kg each go round a circle of radius $R$, under the action of their mutual gravitational attraction. The angular speed of each particle is:
(1) $\sqrt { \frac { G } { 2 R ^ { 3 } } }$
(2) $\frac { 1 } { 2 } \sqrt { \frac { G } { R ^ { 3 } } }$
(3) $\frac { 1 } { 2 R } \sqrt { \frac { 1 } { G } }$
(4) $\sqrt { \frac { 2 G } { R ^ { 3 } } }$

Four particles each of mass $M$, move along a circle of radius $R$ under the action of their mutual gravitational attraction as shown in figure. The speed of each particle is:
(1) $\frac { 1 } { 2 } \sqrt { \frac { G M } { R } (2 \sqrt { 2 } + 1)}$
(2) $\frac { 1 } { 2 } \sqrt { \frac { G M } { R ( 2 \sqrt { 2 } + 1 ) } }$
(3) $\frac { 1 } { 2 } \sqrt { \frac { G M } { R } (2 \sqrt { 2 } - 1)}$
(4) $\sqrt { \frac { G M } { R } }$

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 curve in a level road has a radius 75 m. The maximum speed of a car turning this curved road can be $30 \mathrm{~m~s}^{-1}$ without skidding. If radius of curved road is changed to 48 m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be $\_\_\_\_$ $\mathrm{m~s}^{-1}$.

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 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 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