$A$ 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 $(b)$.

$A$ $1 \,kg$ block situated on a rough incline is connected to a spring of negligible mass having spring constant $100 \,N \,m^{-1}$ as shown in the figure. The block is released from rest with the spring in the unstretched position. The block moves $10 \,cm$ down the incline before coming to rest. The coefficient of friction between the block and the incline is (Take $g=10 \,m \,s^{-2}$ and assume that the pulley is frictionless).

Three forces acting on a body are shown in the figure. To have the resultant force only along the $y-$ direction,the magnitude of the minimum additional force needed is ........... $N$.

$A$ racing car travels on a track (without banking) $ABCDEFA$. $ABC$ is a circular arc of radius $2R$. $CD$ and $FA$ are straight paths of length $R$ and $DEF$ is a circular arc of radius $R = 100 \, m$. The coefficient of friction on the road is $\mu = 0.1$. The maximum speed of the car on straight paths is $50 \, m/s$. Find the minimum time for completing one round.

In the given system,if $M_2 = 2M_1$,the tension in the string is $T$. If the positions of the blocks are interchanged,what will be the new tension in the string?

$A$ block of mass $m$ is moving with a constant acceleration $a$ on a rough plane. If the coefficient of friction between the block and the ground is $\mu$,the power delivered by the external agent after a time $t$ from the beginning is equal to: