A disc is spinning with an angular velocity $? \raisebox{1ex}{$\mathrm{rad}$}\!\left/ \!\raisebox{-1ex}{$\sec $}\right.$ about the axis of spin. The couple applied to the disc causing precession will be

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be

A solid disc of radius $a$ and mass $m$ rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac{2}{\mathit{ }b}g\sin \theta$, where $b$ is _______.  (Round off to the nearest integer)

($g=$ acceleration due to gravity and $\theta =$ angle as shown in figure)

One end of a rod of length $L=1 \mathrm{m}$ is fixed to a point on the circumference of a wheel of radius $R=\frac{1}{\sqrt{3}} \mathrm{m}.$ The other end is sliding freely along a straight channel passing through the center $O$ of the wheel as shown in the figure below. The wheel is rotating with a constant angular velocity $\omega$ (in radian per second) about $O.$

The speed of the sliding end $P$ when $\theta =60°$ is

The angular speed of truck wheel is increased from $900 \mathrm{rpm}$ to $2460 \mathrm{rpm}$ in $26$ seconds. The number of revolutions by the truck engine during this time is ______.

(Assuming the acceleration to be uniform).

A wheel of radius "rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road a small bob of the mud separates from wheel at the highest point and touches the point $B$ on the road as shown in the figure. Then $AB$ is (g-acceleration due to gravity)

A ball is rolling without slipping in a spherical shallow bowl (radius $R$ ) as shown in the figure and is executing simple harmonic motion. If the radius of the ball is doubled, then the time period of oscillation

A sphere, a ring and a disc of same mass and radius are allowed to roll down three similar sufficiently rough inclined planes as shown in the figure from same height.

Which of the following order is true for final $KE$ of the bodies?

A particle of mass$m$ moving horizontally with a velocity $v_0$ collides and sticks to the bottom of the smoothly pivoted hanging rod of mass $M (= 3 m)$and length$l$.

A solid sphere is set into rotation at an angular velocity ${\text{ω}}_0$ and it is then, placed on a rough horizontal surface. The ratio of distances covered by rotational and translational motions up to the start of the pure rolling is (assume uniformly accelerated motion up to start of pure rolling):

A cylinder moves with linear velocity $v$ to the right and angular velocity ω in the anticlockwise direction. The cylinder was initially at rest. The cylinder does not perform pure rolling and stops after some time. What will be the radius of the cylinder?

A sphere of mass $m$ is given some angular velocity about a horizontal axis through its center and gently placed on a plank of mass $m.$ The coefficient of friction between the two is $\mu$. The plank rests on a smooth horizontal surface. The initial acceleration of the center of sphere relative to the plank will be: