$A$ rod of mass $m$ and length $l$ is hinged at one end to a horizontal floor and stands vertically. If it is allowed to fall,the velocity with which its upper end strikes the floor is:

$A$ yo-yo is resting on a rough horizontal table. Forces $F_1, F_2$ and $F_3$ are applied separately as shown. The correct statement is

$A$ uniform circular disc of mass $1.5 \ kg$ and radius $0.5 \ m$ is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5 \ N$ are applied simultaneously along the three sides of an equilateral triangle $XYZ$ with its vertices on the perimeter of the disc (see figure). One second after applying the forces,the angular speed of the disc in $\text{rad } s^{-1}$ is:

$A$ uniform circular disc placed on a rough horizontal surface has initially a velocity $v_0$ and an angular velocity $\omega_0$ as shown in the figure. The disc comes to rest after moving some distance in the direction of motion. Then $\frac{v_0}{r\omega_0}$ is

Consider a body of mass $1.0 \ kg$ at rest at the origin at time $t=0$. $A$ force $\overrightarrow{F}=(\alpha t \hat{i}+\beta \hat{j})$ is applied on the body,where $\alpha=1.0 \ Ns^{-1}$ and $\beta=1.0 \ N$. The torque acting on the body about the origin at time $t=1.0 \ s$ is $\vec{\tau}$. Which of the following statements is (are) true?
$(A)$ $|\vec{\tau}|=\frac{1}{3} \ Nm$
$(B)$ The torque $\vec{\tau}$ is in the direction of the unit vector $+\hat{k}$
$(C)$ The velocity of the body at $t=1 \ s$ is $\overrightarrow{v}=\frac{1}{2}(\hat{i}+2 \hat{j}) \ ms^{-1}$
$(D)$ The magnitude of displacement of the body at $t=1 \ s$ is $\frac{1}{6} \ m$

$A$ hemisphere of mass $3m$ and radius $R$ is free to slide with its base on a smooth horizontal table. $A$ particle of mass $m$ is placed on the top of the hemisphere. If the particle is displaced with a negligible velocity,find the angular velocity of the particle relative to the centre of the hemisphere at an angular displacement $\theta$,when the velocity of the hemisphere is $v$.