Two masses of $0.3 \, kg$ and $0.7 \, kg$ are attached to the ends of a rod of length $1.4 \, m$ and negligible mass. The rod is rotated about an axis perpendicular to its length with a constant angular speed. The point on the rod through which the axis should pass so that the work required to rotate the rod is minimum is:

From a solid sphere of mass $M$ and radius $R$,a cube of maximum possible volume is cut. The moment of inertia of the cube about an axis passing through its center and perpendicular to one of its faces is

$A$ wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. $A$ constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $\tau$ about an axis normal to the plane of the paper passing through the point $Q$. Which of the following options is/are correct?

$A$ circular platform is situated in a horizontal plane and can rotate about a vertical axis passing through its center. $A$ tortoise is sitting at one edge of the platform,and the platform is rotating with a constant angular velocity $\omega_0$. If the tortoise starts moving with a uniform speed along a chord of the circular platform,how will the angular velocity of the platform change with time $t$?

$A$ body of mass $1 \,kg$ is suspended by a weightless string which passes over a pulley of mass $2 \,kg$ as shown in the figure. The mass is released from a height of $1.6 \,m$ from the ground. With what velocity does it strike the ground?

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 $\frac{1}{6} \ m$