$A$ heavy ball of mass $M$ is suspended from the ceiling of a car by a light string of mass $m$ $(m \ll M)$. When the car is at rest,the speed of transverse waves in the string is $60 \, ms^{-1}$. When the car has acceleration $a$,the wave-speed increases to $60.5 \, ms^{-1}$. The value of $a$,in terms of gravitational acceleration $g$,is closest to:

Two masses $m_1 = 5 \ kg$ and $m_2 = 10 \ kg$,connected by an inextensible string over a frictionless pulley,are arranged as shown in the figure. The coefficient of friction of the horizontal surface is $0.15$. The minimum mass $m$ that should be placed on top of $m_2$ to stop the motion is ........ $kg$. (in $.3$)

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 ............ $m/s^2$.

$A$ small block of mass $0.1 \ kg$ lies on a fixed inclined plane $PQ$ which makes an angle $\theta$ with the horizontal. $A$ horizontal force of $1 \ N$ acts on the block through its center of mass as shown in the figure. The block remains stationary if (take $g = 10 \ m/s^2$):

This question has Statement $1$ and Statement $2$. Of the four choices given after the Statements,choose the one that best describes the two Statements.
Statement $1$: If you push on a cart being pulled by a horse so that it does not move,the cart pushes you back with an equal and opposite force.
Statement $2$: The cart does not move because the forces described in Statement $1$ cancel each other.

In the figure,a ball of mass $m$ is tied with two strings of equal length as shown. If the rod is rotated with angular velocity $\omega$,then