$A$ sphere is placed rotating with its centre initially at rest in a corner as shown in figures $(a)$ and $(b)$. The coefficient of friction between all surfaces and the sphere is $\mu = \frac{1}{3}$. Find the ratio of the frictional force $\frac{f_a}{f_b}$ exerted by the ground in situations $(a)$ and $(b)$.

$A$ wooden block of mass $2\; kg$ rests on a soft horizontal floor. When an iron cylinder of mass $25\; kg$ is placed on top of the block,the floor yields steadily and the block and the cylinder together go down with an acceleration of $0.1\; m/s^2$. What is the action of the block on the floor $(a)$ before and $(b)$ after the floor yields? Take $g = 10\; m/s^2$. Identify the action-reaction pairs in the problem.

Place a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \ cm$ and the right one at $90.00 \ cm$. When you attempt to move both fingers slowly towards the center,initially only the left finger slips with respect to the scale and the right finger does not. After some distance,the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_R$ from the center $(50.00 \ cm)$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are $0.40$ and $0.32$,respectively,the value of $x_R$ (in $cm$) is:

$A$ block $B$ is placed on block $A$. The mass of block $B$ is less than the mass of block $A$. Friction exists between the blocks,whereas the ground on which the block $A$ is placed is taken to be smooth. $A$ horizontal force $F$,increasing linearly with time,begins to act on $B$. The acceleration $a_A$ and $a_B$ of blocks $A$ and $B$ respectively are plotted against $t$. The correctly plotted graph is

$A$ force $F=t$ is applied to block $A$ as shown in the figure. The force is applied at $t=0$ seconds when the system was at rest and the string was just straight without tension. Which of the following graphs shows the friction force between block $B$ and the horizontal surface as a function of time $t$?

$A$ helicopter of mass $1000 \;kg$ rises with a vertical acceleration of $15\; m s^{-2}$. The crew and the passengers weigh $300\; kg$. Give the magnitude and direction of the
$(a)$ force on the floor by the crew and passengers,
$(b)$ action of the rotor of the helicopter on the surrounding air,
$(c)$ force on the helicopter due to the surrounding air.