Conservative Force as Gradient of Potential

Q. The potential energy of a body of mass $2\text{kg}$ in a conservative field is $U=(6x-8y)\text{J}$. If the initial velocity of the body is $\overrightarrow{u}=(-1.5\hat{i}+2\hat{j})\text{m/s}$, then the total distance travelled by the particle in the first two seconds is

1. $10\text{m}$
2. $12\text{m}$
3. $15\text{m}$
4. $18\text{m}$

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. 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. Potential energy for a conservative force $\overrightarrow{F}$ is given by $U(x,y)=\cos (x+y)$. Force acting on a particle at position given by coordinates $(0,\frac{\pi }{4})$ is

1. $-\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
2. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
3. $[\frac{1}{2}\hat{i}+\frac{\sqrt{3}}{2}\hat{j}]$
4. $[\frac{1}{2}\hat{i}-\frac{\sqrt{3}}{2}\hat{j}]$

Q. A particle is moving in a circular path of radius $a$ with constant speed under the action of an attractive conservative force. Potential energy of the particle is given by the relation $U=\frac{K}{2r^2}$, where $r$ is the radial distance of the particle from the centre of the circular path. Its total energy will be:

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

Q. The potential energy of $1\text{kg}$ particle moving along the $X-$ axis is given by $U=(\frac{x^4}{4}-\frac{x^2}{2})\text{J}$. The total mechanical energy of the particle is $2\text{J}$. Then maximum speed of the particle is (in $\text{m/s}$)

1. $\frac{3}{\sqrt{2}}$
2. $\frac{1}{\sqrt{2}}$
3. $\sqrt{2}$
4. $2$

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. The potential energy (in joule) of a body of mass $2\text{kg}$ moving in the $x-y$ plane is given by $U=6x+8y$, where the position coordinates $x$ and $y$ are measured in metres. The body is at rest at point $(6\text{m},4\text{m})$ at time $t=0$. It will cross the $y-$ axis at time $t$ equal to

1. $1\text{s}$
2. $2\text{s}$
3. $3\text{s}$
4. $4\text{s}$

Q. The potential energy (in joule) of a body of mass $2\text{kg}$ moving in the $x-y$ plane is given by $U=6x+8y$, where the position coordinates $x$ and $y$ are measured in metres. The body is at rest at point $(6\text{m},4\text{m})$ at time $t=0$. It will cross the $y-$ axis at time $t$ equal to

1. $1\text{s}$
2. $2\text{s}$
3. $3\text{s}$
4. $4\text{s}$

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 particle, which is constrained to move along the x - axis the x - axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as $F\left(x\right)=-kx+a{x}^3$. Here k and a are positive energy U(x) of the particle is

1. 
2. 
3. 
4. 

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. 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. 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. The potential energy of a certain particle is given by $U=\frac{1}{2}(x^2-y^2)$. The force on it is:

1. $-x\hat{i}+y\hat{j}$
2. $-x\hat{i}+z\hat{k}$
3. $-x\hat{i}-y\hat{j}$
4. $x\hat{i}-y\hat{j}$

Q. The potential energy of a body of mass $2\text{kg}$ in a conservative field is $U=(6x-8y)\text{J}$. If the initial velocity of the body is $\overrightarrow{u}=(-1.5\hat{i}+2\hat{j})\text{m/s}$, then the total distance travelled by the particle in the first two seconds is

1. $10\text{m}$
2. $12\text{m}$
3. $15\text{m}$
4. $18\text{m}$

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