$A$ 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:

$A$ body of mass $2 \ kg$ is initially at rest. It moves on a table under the influence of a horizontal force of $7 \ N$. If the coefficient of kinetic friction is $0.1$,then the work done by the applied force and the work done by the frictional force in $10 \ s$ are respectively:

Work done by a frictional force is

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:

Match the items in Column-$I$ with those in Column-$II$.

Column-$I$	Column-$II$
$(1)$ Newton's third law of motion	$(a)$ $\vec{F}_{12} = -\vec{F}_{21}$
$(2)$ Law of conservation of momentum	$(b)$ $\Delta \vec{p} = 0$
	$(c)$ $\vec{F}_{12} = \vec{F}_{21}$

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