$A$ boy whose mass is $50\,kg$ stands on a spring balance inside a lift. The lift starts to ascend with an acceleration of $2\,m/s^2$. The reading of the machine or balance $(g = 10\,m/s^2)$ is ........ $kg$.

$A$ frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. $A$ coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force $\vec{F}_{\text{rot}}$ experienced by a particle of mass $m$ moving on the rotating disc and the force $\vec{F}_{\text{in}}$ experienced by the particle in an inertial frame of reference is $\vec{F}_{\text{rot}} = \vec{F}_{\text{in}} + 2m(\vec{v}_{\text{rot}} \times \vec{\omega}) + m(\vec{\omega} \times \vec{r}) \times \vec{\omega}$,where $\vec{v}_{\text{rot}}$ is the velocity of the particle in the rotating frame of reference and $\vec{r}$ is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius $R$ rotating counter-clockwise with a constant angular speed $\omega$ about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc,the $x$-axis along the slot,the $y$-axis perpendicular to the slot and the $z$-axis along the rotation axis $(\vec{\omega} = \omega \hat{k})$. $A$ small block of mass $m$ is gently placed in the slot at $\vec{r} = (R/2) \hat{i}$ at $t = 0$ and is constrained to move only along the slot.
$(1)$ The distance $r$ of the block at time $t$ is:
$(A)$ $\frac{R}{4}(e^{\omega t} + e^{-\omega t})$
$(B)$ $\frac{R}{2} \cos \omega t$
$(C)$ $\frac{R}{4}(e^{2\omega t} + e^{-2\omega t})$
$(D)$ $\frac{R}{2} \cos 2\omega t$
$(2)$ The net reaction of the disc on the block is:
$(A)$ $\frac{1}{2} m \omega^2 R(e^{\omega t} - e^{-\omega t}) \hat{j} + mg \hat{k}$
$(B)$ $\frac{1}{2} m \omega^2 R(e^{\omega t} + e^{-\omega t}) \hat{j} + mg \hat{k}$
$(C)$ $-m \omega^2 R \cos \omega t \hat{j} - mg \hat{k}$
$(D)$ $m \omega^2 R \sin \omega t \hat{j} - mg \hat{k}$

$A$ solid sphere of mass $2\, kg$ is resting inside a cube as shown in the figure. The cube is moving with a velocity $\vec v = (5t\,\hat i + 2t\,\hat j)\,m/s$. Here $t$ is the time in seconds. All surfaces are smooth. The sphere is at rest with respect to the cube. Find the total force $N$ exerted by the sphere on the cube. (Take $g = 10\, m/s^2$)

$A$ man has a mass of $80 \text{ kg}$. He stands on a weighing scale in a lift which is moving upwards with a uniform acceleration of $6 \text{ m/s}^2$. What would be his apparent weight in $\text{kg}$? (Take $g = 10 \text{ m/s}^2$)

$A$ ball is suspended on a thread from the ceiling of a car. The brakes are applied and the speed of the car changes from $5\, m/s$ to $\frac{5}{3}\, m/s$ during the time interval of $3\, s$. The angle that the thread will deviate from the vertical in the new equilibrium position is:

$A$ plank is moving in a horizontal direction with a constant acceleration $a \hat{i}$. $A$ uniform rough cubical block of side $l$ rests on the plank and is at rest relative to the plank. Let the centre of mass of the block be at $(0, l/2)$ at a given instant. If $a = g/10$,then the normal reaction exerted by the plank on the block at that instant acts at