$A$ particle at rest starts moving with a constant angular acceleration of $4 \ rad/s^2$ in a circular path. The time at which the magnitudes of its centripetal acceleration and tangential acceleration will be equal is (in seconds):

The velocity $\vec{v}$ of a particle of mass $m$ acted upon by a constant force is given by $\vec{v}(t) = A[\cos(kt) \hat{i} - \sin(kt) \hat{j}]$. Then the angle between the force and the velocity of the particle is (Here $A$ and $k$ are constants). (in $^{\circ}$)

If the frequency of an object in uniform circular motion is doubled,its acceleration becomes

If the linear speed of a particle performing uniform circular motion is increased to $4$ times and its angular velocity is doubled,then the centripetal acceleration of the particle becomes .....

$A$ flywheel starts from rest and rotates with a constant angular acceleration of $3.0 \ rad/s^2$. An observer notes that it covers an angle of $120 \ rad$ in a time interval of $4.0 \ s$. How long had the wheel been rotating before the observer started the observation?

Three particles $A, B$ and $C$ move in a circle of radius $r = \frac{1}{\pi} \, m$ in the anticlockwise direction with speeds $1 \, m/s$,$2.5 \, m/s$ and $2 \, m/s$ respectively. The initial positions of $A, B$ and $C$ are as shown in the figure. The ratio of the distance travelled by $B$ and $C$ by the instant $A, B$ and $C$ meet for the first time is