The upper half of an inclined plane of inclination $\theta$ is perfectly smooth,while the lower half is rough. $A$ block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and the lower half of the plane is given by:

$A$ block of mass $5\,kg$ starting from rest is pulled up on a smooth inclined plane making an angle of $30^{\circ}$ with the horizontal with an effective acceleration of $1\,m/s^2$. The power delivered by the pulling force at $t = 10\,s$ from the start is $.....\,W$. [Use $g = 10\,m/s^2$] (calculate the nearest integer value).

$A$ block $B$,lying on a table,has weight $w$. The coefficient of static friction between the block and the table is $\mu$. Assume that the cord between $B$ and the knot is horizontal. The maximum weight of the block $A$ for which the system will be stationary is

$A$ $100 \, kg$ gun fires a ball of $1 \, kg$ horizontally from a cliff of height $500 \, m$. It falls on the ground at a distance of $400 \, m$ from the bottom of the cliff. Find the recoil velocity of the gun. (Acceleration due to gravity $g = 10 \, m/s^2$) (in $, m/s$)

$A$ $1.0 \ kg$ block of wood sits on top of an identical block of wood,which sits on top of a flat level table made of plastic. The coefficient of static friction between the wood surfaces is $\mu_1$,and the coefficient of static friction between the wood and plastic is $\mu_2$. $A$ horizontal force $F$ is applied to the top block only,and this force is increased until the top block starts to move. The bottom block will move with the top block if and only if

$A$ force $F$ is applied on the top of a cube as shown in the figure. The coefficient of friction between the cube and the ground is $\mu$. If $F$ is gradually increased,the cube will topple before sliding. Then the range of $\mu$ is: