Exercise-4

A body is executing simple harmonic motion. At a displacement $x$, its potential energy is $E_1$ and at a displacement $y$, its potential energy is $E_2$. The potential energy $E$ at a displacement $(x+y)$ is

A stone of mass $1 \mathrm{kg}$ tied to a light inextensible string of length $L=\frac{10}{3} \mathrm{m}$, is whirling in a circular path in a vertical plane. The ratio of maximum tension in the string to the minimum tension in the string is $4.$ If $g$ is taken to be $10 \mathrm{m} {\mathrm{s}}^{-2}$, the speed of the stone at the highest point of the circle is:

A particle is moving in a vertical circle. The tensions in the string when passing through two positions at angles $30°$ and $60°$ from vertical (lowest position) are $T_1$ and $T_2$ respectively then

A car of mass $m$ starts moving so that its velocity varies according to the law $v=\beta \sqrt{s},$ where $\beta$ is a constant, and $s$ is the distance covered. The total work performed by all the forces which are acting on the car during the first $t$ seconds after the beginning of motion is

The graph between kinetic energy $E$ and velocity $v$ is :

The work done on a particle of mass $m$ by a force $K\left[\frac{x}{{\left(x^2+y^2\right)}^{{\displaystyle \frac{3}{2}}}}\mathrm{i}+\frac{y}{{\left(x^2+y^2\right)}^{{\displaystyle \frac{3}{2}}}}\mathrm{j}\right]$ ($K$ being a constant of appropriate dimensions), when the particle is taken from the point $\left(a, 0\right)$ to the point $\left(0, a\right)$ along a circular path of radius $a$ about the origin in the $X-Y$ plane is:

A toy car of mass $5 \mathrm{kg}$ moves up a rampíunder the influence of force $F$ plotted against displacement $x.$ The maximum height attained is given by

A particle is taken from point $A$ to point $B$ under the influence of a force field. Now it is taken back from $B$ to $A$ and it is observed that the work done in taking the particle from $A$ to $B$ is not equal to the work done in taking it from $B$ to $A.$ If $W_{nc}$ and $W_c$ is the work done by non-conservative forces and conservative forces present in the system respectively, $\Delta U$ is the change in potential energy, $\Delta K$ is the change in kinetic energy, then