Conservative Force as Gradient of Potential

Q. The figure below shows a graph of potential energy $U\left(x\right)$ verses position $x$ for a particle executing one dimentional motion along the x-axis. The total mechanical energy of the system is indicated by the dashed line. Choose the correct statement for the position of particle varying between points $A$ and $G$.

1. The magnitude of force is maximum at $D$
2. The kinetic energy is maximum at $B$
3. The velocity is zero at $A$ and $G$
4. None of the above

Q. A particle of mass $\frac{10}{7}\text{kg}$ is moving along the positive $x$ direction. Its initial position is $x=0$ and initial velocity is $1\text{m/s}$. The graph representing the power delivered the by force vs distance travelled for the straight line motion of the particle is given below.


The velocity of particle at $x=10\text{m}$ is:

1. $\frac{100}{3}\text{m/s}$
2. $2\text{m/s}$
3. $4\text{m/s}$
4. $3\sqrt{2}\text{m/s}$

Q. Potential energy $U\left(x\right)$ and associated force $F\left(x\right)$ bear the relation $F\left(x\right)=\frac{-dU\left(x\right)}{dx}$. Dependence of potential energy of a two-particle system on the separation $x$ between them is shown in the following figure.


Which of the following graphs shows the correct variation of force $F\left(x\right)$ with $x$?

1. 
2. 
3. 
4. 

Q. The potential energy function of a particle is given by $U=-(x^2+y^2+z^2)\text{J}$, where $x,y$ and $z$ are in meters. Find the force acting on the particle at point $A(1\text{m},3\text{m},5\text{m})$.

1. $\sqrt{8}\text{N}$
2. $\sqrt{136}\text{N}$
3. $\sqrt{140}\text{N}$
4. $10\sqrt{14}\text{N}$

Q. A particle is released from rest at origin. It moves under the influence of a potential field, $U=x^2-3x$. Find the Kinetic energy of the particle at $x=2$.

1. $2\text{J}$
2. $1\text{J}$
3. $1.5\text{J}$
4. $0\text{J}$

Q. The potential energy for a conservative force system is given by $U=a{x}^3-bx$, where $a$ and $b$ are constants. Choose from the options, the correct $x$ coordinate(s) of the equilibrium position(s).

1. $+\sqrt{\frac{b}{2a}}$
2. $-\sqrt{\frac{b}{2a}}$
3. $+\sqrt{\frac{b}{3a}}$
4. $-\sqrt{\frac{b}{3a}}$

Q. A plot of potential energy function $U\left(x\right)=k{x}^2$, where $x$ is the displacement and $k=$ constant is shown below. The correct conservative force function $F\left(x\right)$ is

1. 
2. 
3. 
4. 

Q. A particle of mass m is moving in a horizontal circle of radius r, under a centripetal force equal to $\left(\frac{-k}{r^2}\right)$ where k is constant. What is the total energy of the particle?

1. -$\frac{K}{2r}$
2. -$\frac{K}{2r^2}$
3. –$\frac{K^2}{2r}$
4. -$\frac{K}{r}$

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

1. The particle has speed $20\text{m/s}$ at $t=4\text{sec}$
2. The particle has an acceleration $25{\text{m/s}}^2$
3. The acceleration of the particle is normal to the initial velocity
4. None of the above are correct

Q. A particle is moving in a circular path of radius '$a$' under the action of an attractive potential, $U=-\frac{k}{2r^2}$ (where $r$ is the radial distance). Its total energy is

1. Zero
2. $-\frac{3}{2}\frac{k}{a^2}$
3. $-\frac{k}{4a^2}$
4. $\frac{k}{2a^2}$

Q. A particle has potential energy dependent on its position on the $x$ axis, represented by the function $U\left(x\right)=e^{2x}+1$ for all real values of $x$, where $U\left(x\right)$ and $x$ are given in standard units. The force it feels at position $x=1$ is closest to
[Take $e=2.72$]

1. $8.39\text{N}$
2. $-8.39\text{N}$
3. $14.8\text{N}$
4. $-14.8\text{N}$

Q. The potential energy $U$ in joules of a particle of mass $1\text{kg}$ moving in the $x-y$ plane obeys the law $U=3x+4y$, where $(x,y)$ are the co-ordinates of the particle in metres. If the particle is released from rest at $(6,4)$ at time $t=0$, then

1. the particle has zero acceleration
2. co-ordinates of the particle at $t=1\text{s}$ is $(2,4.5)$
3. co-ordinates of the particle at $t=1\text{s}$ is $(4.5,2)$
4. co-ordinates of the particle at $t=1\text{s}$ is $(2,2)$

Q. A particle of mass m is moving in a horizontal circle of radius r, under a centripetal force equal to $\left(\frac{-k}{r^2}\right)$ where k is constant. What is the total energy of the particle?

1. -$\frac{K}{2r}$
2. -$\frac{K}{2r^2}$
3. –$\frac{K^2}{2r}$
4. -$\frac{K}{r}$

Q. The potential energy function $U\left(x\right)$ of a particle moving along $x$ direction is given by $U\left(x\right)=a{x}^2-bx$. Find the equilibirum point $\left(x_e\right)$.

1. $x_e=\frac{a}{2b}$
2. $x_e=\frac{b}{2a}$
3. $x_e=\frac{a^2}{b^2}$
4. $x_e=\frac{2a}{b}$

Q. A particle is moving in a circular path of radius '$a$' under the action of an attractive potential, $U=-\frac{k}{2r^2}$ (where $r$ is the radial distance). Its total energy is

1. Zero
2. $-\frac{3}{2}\frac{k}{a^2}$
3. $-\frac{k}{4a^2}$
4. $\frac{k}{2a^2}$