$A$ body falls freely under gravity. Its speed is $v$ when it has lost an amount $U$ of the gravitational potential energy. Then its mass is

$A$ demonstration apparatus on a table in the lab is shown in the diagram. It consists of a metal track (shown as a thick solid line in the figure) along which a perfectly spherical marble can roll without slipping. In one run,the marble is released from rest at a height $h$ above the table on the left section,rolls down one side and then up the other side without slipping,briefly stopping when it has reached $h_1$. Assuming the table to be horizontal and neglecting air drag as well as any energy loss due to rolling,which of the following is true?

$A$ particle of mass $m \ kg$ moves along the $X$-axis with its velocity varying with the distance travelled as $v=k x^\beta$, where $k$ is a positive constant. The total work done by all the forces during displacement of the particle from $x=0$ to $x=d$ is close to

The bob of a pendulum was released from a horizontal position. The length of the pendulum is $10 \ m$. If it dissipates $10 \%$ of its initial energy against air resistance,the speed with which the bob arrives at the lowest point is : [Use $g = 10 \ ms^{-2}$]

State and establish the principle of conservation of mechanical energy.

$A$ particle $(m = 1 \; kg)$ slides down a frictionless track $(AOC)$ starting from rest at a point $A$ (height $2 \; m$). After reaching $C$,the particle continues to move freely in air as a projectile. When it reaches its highest point $P$ (height $1 \; m$),the kinetic energy of the particle (in $J$) is: (Figure drawn is schematic and not to scale; take $g = 10 \; ms^{-2}$)