A solid sphere of mass $1 \mathrm{kg}$ and radius $1 \mathrm{m}$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta =30°$ from the horizontal. Two forces of magnitude $1 \mathrm{N}$ each, parallel to the incline, act on the sphere, both at distance $r=0.5 \mathrm{m}$ from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is _____ $\mathrm{m} {\mathrm{s}}^{-2}$. (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$).

The machine as shown has $2$ rods of length $1 \mathrm{m}$ connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has roller that rolls along the floor in a slot. As the roller goes back and forth, a $2 \mathrm{kg}$ weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a :

Calculate the acceleration of the block and trolly system shown in the figure. The coefficient of kinetic friction between the trolly and the surface is $0.05.$ ($g=10 \mathrm{m} {\mathrm{s}}^{-2},$ mass of the string is negligible and no other friction exists).

A solid sphere of mass $1 \mathrm{kg}$ and radius $1 \mathrm{m}$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta =30°$ from the horizontal. Two forces of magnitude $1 \mathrm{N}$ each, parallel to the incline, act on the sphere, both at distance $r=0.5 \mathrm{m}$ from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is $\mathrm{m} {\mathrm{s}}^{-2}$. (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

Two blocks $A$ and $B$ of masses $1.5 \mathrm{kg}$ and $0.5 \mathrm{kg}$ respectively are connected by a massless inextensible string passing over a frictionless pulley as shown in the figure. Block A is lifted until block B touches the ground and then block A is released. The initial height of block A is $80 \mathrm{cm}$ when block B just touches the ground. The maximum height reached by block $\mathrm{B}$ from the ground after the block A falls on the ground is

Two inclined planes are placed as shown in figure.
A block is projected from the Point $A$ of inclined plane $AB$ along its surface with a velocity just sufficient to carry it to the top Point $B$ at a height $10 \mathrm{m}$. After reaching the Point $B$ the block slides down on inclined plane $BC$. Time it takes to reach to the point $C$ from point $A$ is $t\left(\sqrt{2}+1\right)\mathrm{s}$. The value of t is _____ (use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)