A block of mass $M$ is being pulled along rough horizontal surface. The coefficient of friction between the block and the surface is $\mu $. If another block of mass $M/2$ is placed on the block and it is again pulled on the surface, the coefficient of friction between the block and the surface will be

A block of mass $m$ is on an inclined plane of angle $\theta$. The coefficient of friction between the block and the plane is $\mu$ and $\tan \theta>\mu$. The block is held stationary by applying a force $\mathrm{P}$ parallel to the plane. The direction of force pointing up the plane is taken to be positive. As $\mathrm{P}$ is varied from $\mathrm{P}_1=$ $m g(\sin \theta-\mu \cos \theta)$ to $P_2=m g(\sin \theta+\mu \cos \theta)$, the frictional force $f$ versus $P$ graph will look like

A circular racetrack of radius $300\; m$ is banked at an angle of $15^o$. If the coefficient of friction between the wheels of a race-car and the road is $0.2$, what is the

$(a)$ optimum speed of the racecar to avoid wear and tear on its tyres, and

$(b)$ maximum permissible speed to avoid slipping ?

A uniform rope lies on a horizontal table so that a part of it hangs over the edge. The rope begins to slide down when the length of the hanging part is $25\%$ of the entire length. The coefficient of friction between the rope and the table is

In the figure shown, a block of weight $10 \,N$ resting on a horizontal surface. The coefficient of static friction between the block and the surface ${\mu _s} = 0.4$. A force of $3.5\, N$ will keep the block in uniform motion, once it has been set in motion. A horizontal force of $3 \,N$ is applied to the block, then the block will

A block of mass $m$ is in contact with the cart $C$ as shown in the figure. The coefficient of static friction between the block and the cart is $\mu .$ The acceleration $\alpha$ of the cart that will prevent the block from falling satisfies