$A$ uniform rod $AB$ of weight $100 \, N$ rests on a rough peg at $C$ and a force $F$ acts at $A$ as shown in the figure. If $BC = CM$ and $\tan \alpha = 4/3$,the minimum coefficient of friction at $C$ is:

$A$ long plank $P$ of mass $5\, kg$ is placed on a smooth floor. On $P$ is placed a block $Q$ of mass $2\, kg$. The coefficient of friction between $P$ and $Q$ is $0.5$. If a horizontal force $15\, N$ is applied to $Q$,as shown,and you may take $g$ as $10\, N/kg$,then which of the following statements is correct?

$A$ particle initially at rest is subjected to two forces. One is constant,the other is a retarding force proportional to the particle velocity. In the subsequent motion of the particle,

$A$ 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:

$A$ balloon of mass $M$ is descending at a constant acceleration $\alpha$. When a mass $m$ is released from the balloon,it starts rising with the same acceleration $\alpha$. Assuming that its volume does not change,what is the value of $m$?

At the moment $t=0$, a time-dependent force $F=at$ (where $a$ is a constant equal to $1 \,Ns^{-1}$) is applied to a body of mass $1 \,kg$ resting on a smooth horizontal plane as shown in the figure. If the direction of this force makes an angle $45^{\circ}$ with the horizontal, then the velocity of the body at the moment it leaves the plane is (acceleration due to gravity $=10 \,ms^{-2}$)