$A$ uniform rope of total length $l$ is at rest on a table with a fraction $f$ of its length hanging (see figure). If the coefficient of friction between the table and the rope is $\mu$,then:

$A$ block of mass $10\, kg$ starts sliding on a surface with an initial velocity of $9.8\, m/s$. The coefficient of friction between the surface and the block is $0.5$. The distance covered by the block before coming to rest is: [use $g = 9.8\, m/s^2$].........$m$

$A$ movable steel plate is placed between fixed steel and brass plates and the stack of plates is subjected to a weight of $100 \ N$ as shown in the figure. The coefficient of kinetic friction for steel on steel is $0.57$ and for steel on brass is $0.44$. Assuming that the entire weight comes onto the stack and that the weight of the plates is negligible in comparison to the applied weight, the force required to move the middle plate (in $N$) is

Consider a block and trolley system as shown in the figure. If the coefficient of kinetic friction between the trolley and the surface is $0.04$,the acceleration of the system in $m/s^2$ is (Consider that the string is massless and unstretchable and the pulley is also massless and frictionless):

$A$ body of mass $1 \,kg$ is moving with a velocity $10 \,ms^{-1}$ on a horizontal rough surface having a coefficient of kinetic friction $0.4$. If the constant force is removed, the body comes to rest in a time (Acceleration due to gravity $g = 10 \,ms^{-2}$). (in $\,s$)

$A$ $300 \ kg$ crate is dropped vertically onto a conveyor belt that is moving at $1.20 \ m/s$. $A$ motor maintains the belt's constant speed. The belt initially slides under the crate,with a coefficient of friction of $0.400$. After a short time,the crate is moving at the speed of the belt. During the period in which the crate is being accelerated,find the work done by the motor which drives the belt.