Uniform circular motion

Consider a large disk of radius $R$ and two smaller disks, each of radius $r=R/50$, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation $\Delta\theta$ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities $\omega$ and $2\omega$ while the large disk is held stationary. The time $\tau$ at which the smaller disks are again in contact is:
[Use $\sin(\Delta\theta)=\Delta\theta$ and ignore gravity.]

A particle is rotating in a circular path and at any instant its motion can be described as $\theta=\dfrac{5t^4}{40}-\dfrac{t^3}3$. The angular acceleration of the particle after 10 seconds is ________ rad/s².

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in Hz is _____.

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $\mathrm{m\,s^{-1}}$) at time $t = \dfrac\pi3$ s.

Which one of the following options is correct?

For a particle in uniform circular motion, the acceleration $\vec a$ at any point $P(R,\theta)$ on the circular path of radius $R$ is (when $\theta$ is measured from the positive $x$-axis and $v$ is uniform speed) :

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

In the $xy$-plane, the region $y > 0$ has a uniform magnetic field $B_1\hat k$ and the region $y < 0$ has another uniform magnetic field $B_2\hat k$. A positively charged particle is projected from the origin along the positive $y$-axis with speed $v_0 = \pi\ \mathrm{m\,s^{-1}}$ at $t = 0$, as shown in the figure. Neglect gravity in this problem. Let $t = T$ be the time when the particle crosses the $x$-axis from below for the first time. If $B_2 = 4B_1$, the average speed of the particle, in $\mathrm{m\,s^{-1}}$, along the $x$-axis in the time interval $T$ is ______.

The distance $r$ of the block at time $t$ is

Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute(rpm) to ensure proper mixing is close to :

(Take the radius of the drum to be 1.25 m and its axle to be horizontal) :

If a body moving in a circular path maintains constant speed of 10 ms$^{-1}$, then which of the following correctly describes relation between acceleration and radius ?

A particle is moving in a circular path of radius a, with a constant velocity $v$ as shown in the figure. The center of circle is marked by 'C'. The angular momentum from the origin O can be written as :

Two identical discs of same radius $R$ are rotating about their axes in opposite directions with the same constant angular speed $\omega$. The discs are in the same horizontal plane. At time $t=0$, the points $P$ and $Q$ are facing each other as shown in the figure. The relative speed between the two points $P$ and $Q$ is $v_r$. In one time period $(T)$ of rotation of the discs, $v_r$ as a function of time is best represented by

Two small particles of equal masses start moving in opposite directions from a point A in a horizontal circular orbit. Their tangential velocities are $v$ and $2v$, respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at A, these two particles will again reach the point A?

A particle moves in the X-Y plane under the influence of a force such that its linear momentum is $\vec p(t)=A\left[\hat i\cos(kt)-\hat j\sin(kt)\right]$, where $A$ and $k$ are constants. The angle between the force and the momentum is