$A$ man of mass $80\, kg$ is standing in an elevator which is moving with an acceleration of $6\, m/s^2$ in the upward direction. The apparent weight of the man will be ......... $N$ $(g = 10\, m/s^2)$.

$A$ container of mass $m$ is pulled by a constant force $F$. Inside the container,a second block of the same mass $m$ is placed,connected to the wall by a massless spring of constant $k$. Initially,the spring is at its natural length. Find the velocity of the container at the instant the compression in the spring is maximum for the first time.

$A$ body of mass $4 \, kg$ weighs $4.8 \, kg$ when suspended in a moving lift. The acceleration of the lift is

$A$ 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 $5/3 \, m/s$ during a time interval of $3 \, s$. Find the angle $\theta$ that the thread will deviate from the vertical.

$A$ 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 $g = 10\, m/s^2$)

$A$ piece of wire is bent in the shape of a parabola $y=kx^2$ ($y$-axis vertical) with a bead of mass $m$ on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the $x$-axis with a constant acceleration $a$. The distance of the new equilibrium position of the bead,where the bead can stay at rest with respect to the wire,from the $y$-axis is