Kinetic and Potential Energy

 A block is attached to the end of an ideal spring and moved from coordinate $x_i$ to coordinate $x_f.$ The relaxed position is at $x=0.$ The work done by spring is positive if

If the kinetic energy of a particle continuously increases with time then?

A block of mass $250 \mathrm{g}$ is kept on a vertical spring of spring constant $100 \mathrm{N} {\mathrm{m}}^{-1}$ fixed from below. The spring is now compressed to have a length $10 \mathrm{cm}$ shorter than its natural length and the system is released from this position. How high does the block rise? Take $\mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}.$

The displacement of an object of mass $3 \mathrm{kg}$ is given by the relation $S=\frac{1}{3}{t}^2$ , where $\mathrm{t}$ is time in seconds. If the work done by the net force on the object in $2 \mathrm{s}$ is $\frac{p}{q} \mathrm{joule}$, where $p$ and $q$ are smallest integer values, then what is the value of $p+q$?

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

The potential energy of a $2 \mathrm{kg}$ particle, free to move along the $x$ -axis is given by $V\left(x\right)=\left(\frac{x^4}{4}-\frac{x^2}{2}\right) \mathrm{J}.$ The total mechanical energy of the particle is $2 \mathrm{J}$ then, the maximum speed (in ${\mathrm{ms}}^{-1}$) is

Kinetic energy of a particle moving in a straight line varies with time as $K=4t^2$. The force acting on the particle

A block of mass $5 \mathrm{kg}$ is released from rest when compression in spring is $2 \mathrm{m}$. The block is not attached with the spring and the natural length of the spring is $4 \mathrm{m}$. Maximum height of the block from ground is: $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$,

A ball is thrown vertically downwards from a height of $20 \mathrm{m}$ with an initial velocity ${\mathrm{v}}_0$ . It collides with the ground, loses $50$ percent of its energy in a collision and rebounds to the same height. The initial velocity ${\mathrm{v}}_0$ is (Take $\mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}$ )

Consider a vertical tube open at both ends. The tube consists of two parts, each of different cross-sections and each part having a piston which can move smoothly in respective tubes. The two pistons are joined together by an inextensible wire. The combined mass of the two pistons is $5 \mathrm{kg}$and area of cross-section of the upper piston is $10 {\mathrm{cm}}^2$greater than that of the lower piston. The amount of gas enclosed by the pistons is one mole. When the gas is heated slowly, the pistons moves by $50 \mathrm{cm}$ as shown in figure. The rise in temperature of the gas is in the form $\frac{X}{R} \mathrm{K}$, where $R$ is universal gas constant. Use, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and outside pressure$={10}^5 \mathrm{N} {\mathrm{m}}^{-2}$. Fill value of $X$.

Two blocks of masses $m$ and $M$ are moving with speeds $v_1$ and $v_2$ $({\mathrm{v}}_1>{\mathrm{v}}_2)$ in the same direction on the frictionless surface respectively, $M$ being ahead of $m$. An ideal spring of force constant $k$ is attached to the backside of $M$ (as shown). The maximum compression of the spring when the block collides is

A $10 \mathrm{kg}$ small block is pulled in the vertical plane along a frictionless surface in the form of an arc of a circle of radius $10 \mathrm{m}$. The applied force is of $200 \mathrm{N}$ as shown in the figure. If the block started from rest at $A$, the speed at $B$ would be: $(g = 10 \mathrm{m} {\mathrm{s}}^{-2})$

A block of mass $m$ is placed inside a smooth hollow cylinder of radius $R$ whose axis is kept horizontally. Initially, the system was at rest. Now cylinder is given constant acceleration $2\mathit{ }g$ in the horizontal direction by an external agent. The maximum angular displacement of the block with the vertical is: 

Which of the following graph may represent the speed of block as a function of distance $s$ from the wall $w$ , when an iron block of mass $m$ resting in contact with a support $S$ is released when it has compressed the ideal spring on a smooth horizontal floor as shown. The block is not attached with the support $S$.

One of the forces acting on a particle is conservative then which of the following statement(s) are true about this conservative force?

If the given plot shows the variation of $U$, the potential energy of interaction between two particles with the distance separating them is $r.$ Then, which of the following statements are correct.

The potential energy of a particle varies with distance $x$ as shown in the graph.

If the zero of the potential energy is to be chosen at the point $\left(2, 2, 2\right)$, then the potential energy for the force $\overrightarrow{\mathit{F}}=yz\hat{\mathrm{i}}+xz\hat{\mathrm{j}}+xy\hat{\mathrm{k}}$ is

A particle of mass $5 \mathrm{kg}$ moving in the $x\mathit{-}y$ plane has its potential energy given by $U=\left(-7x+24y\right) \mathrm{J}$. The particle is initially at origin and has a velocity $\overrightarrow{u}=\left(14.4\hat{\mathrm{i}}+4.2\hat{\mathrm{j}}\right)$. Then,

Potential energy function along $x-$axis, in a certain force field, is given as $U\left(x\right)=\frac{x^4}{4}-2x^3+\frac{11}{2}{x}^3-6x$. For the given force field,

$\left(\mathrm{i}\right)$ the points of equilibrium are $x=1, x=2$ and $x=3$.

$\left(\mathrm{ii}\right)$ the point $x=2$ is a point of unstable equilibrium.

$\left(\mathrm{iii}\right)$ the points $x=1$ and $x=3$ are points of stable equilibrium.

$\left(\mathrm{iv}\right)$ there exists no point of neutral equilibrium.

Which of the above statements is correct?