$A$ man is standing in a lift which goes up and comes down with the same constant acceleration. If the ratio of the apparent weights in the two cases is $2 : 1$,then the acceleration of the lift is ......... $m/s^2$.

$A$ simple pendulum is set up in a trolley which moves to the right with an acceleration $a$ on a horizontal plane. Then the thread of the pendulum in the mean position makes an angle $\theta$ with the vertical.

$A$ circular table is rotating with an angular velocity of $\omega \text{ rad/s}$ about its axis (see figure). There is a smooth groove along a radial direction on the table. $A$ steel ball is gently placed at a distance of $1 \text{ m}$ on the groove. All the surfaces are smooth. If the radius of the table is $3 \text{ m}$,the radial velocity of the ball with respect to the table at the time the ball leaves the table is $x \sqrt{2} \omega \text{ m/s}$,where the value of $x$ is............

$A$ person is standing in an elevator. In which of the following situations does he find his weight to be less than his actual weight?

$A$ trolley is accelerating down an incline of angle $\theta$ with acceleration $g \sin \theta$. Which of the following is correct? ($\alpha$ is the constant angle made by the string with the vertical.)

The free surface of oil in a tanker,at rest,is horizontal. If the tanker starts accelerating,the free surface will be tilted by an angle $\theta$. If the acceleration is $a \ m/s^2$,what will be the slope of the free surface?