$A$ system consists of three masses $m_1$,$m_2$,and $m_3$ connected by a string passing over a pulley $P$. The mass $m_1$ hangs freely,and $m_2$ and $m_3$ are on a rough horizontal table (the coefficient of friction $= \mu$). The pulley is frictionless and of negligible mass. Find the downward acceleration of mass $m_1$. (Assume $m_1 = m_2 = m_3 = m$)

$A$ body of mass $2 \,kg$ is acted upon by two forces each of magnitude $1 \,N$, making an angle of $60^{\circ}$ with each other. The net acceleration of the body (in $m/s^2$) is

$A$ cubical box of side $a$ sitting on a rough table-top is pushed horizontally with a gradually increasing force until the box moves. If the force is applied at a height from the table top which is greater than a critical height $H$,the box topples first. If it is applied at a height less than $H$,the box starts sliding first. Then,the coefficient of friction between the box and the table top is

Five persons $A, B, C, D$ and $E$ are pulling a cart of mass $100 \, kg$ on a smooth surface and the cart is moving with an acceleration of $3 \, m/s^2$ in the east direction. When person $A$ stops pulling,it moves with an acceleration of $1 \, m/s^2$ in the west direction. When person $B$ stops pulling,it moves with an acceleration of $24 \, m/s^2$ in the north direction. The magnitude of the acceleration of the cart when only $A$ and $B$ pull the cart,keeping their directions the same as the original directions,is ............ $m/s^2$.

Consider the following statements $A$ and $B$ and identify the correct answer given below:
$(A)$ $A$ body initially at rest is acted upon by a constant force. The rate of change of its kinetic energy varies linearly with time.
$(B)$ When a body is at rest, it must be in equilibrium.

$A$ $10 \, kg$ ball attached to the end of a rigid massless rod of length $L = 1 \, m$ rotates at a constant speed in a horizontal circle of radius $r = 0.5 \, m$ and period $T = 1.57 \, s$ as shown in the figure. The force exerted by the rod on the ball is ........ $N$.