Conservative and Non Conservative Forces

Q.

The potential energy of a particle varies with distance $x$ from a fixed origin as $U=\frac{\left(A\sqrt{x}\right)}{(x^2+B)}$, where $A$and $B$ are the dimensional constants, then find the dimensional formula for $AB$.

Q.

The negative of the space rate of change of potential energy is equal to

1. force acting on the particle in the direction of displacement

2. acceleration of the particle, perpendicular to displacement

3. power

4. impulse

Q.

A particle of mass m is moving in a horizontal circle of radius r under a centripetal force equal to $\frac{-K}{r^2}$ , where K is a constant. The total energy of the particle is

1. $\frac{K}{2r}$

2. $-\frac{K}{2r}$

3. $-\frac{K}{r}$

4. $\frac{K}{r}$

Q. If $U_1,U_2$ represents "change in potential energy" of a body with mass $m$ moving from $A$ to $B$ along two different paths $1$, & $2$ (as shown in the figure) in the gravitational field. Then

1. $U_1=U_2$
2. $U_1>U_2$
3. $U_1<U_2$
4. $U_1\ge U_2$

Q. Friction is a conservative force.

1. False
2. True

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

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

Q. The potential energy of a conservative system is given by $U=A{x}^2-Bx$, where $x$ represents the position of the particle. $A$ and $B$ are positive constants. Then

1. Force acting on the system will be $(B-2Ax)$
2. At equilibrium, potential energy will be $\frac{-B^2}{4A}$
3. system is in stable equilibrium.
4. None

Q. A point like object of mass M is thrown up from point A as shown in the figure. It slides along the full length of the smooth track ABC (of radius R for part BC). Let $R=1m,AB=2m,M=0.5kg,g=10m/s^2\&OD=3m.$

1. Minimum speed $V_0$ to slide full length of the track is $\sqrt{60}m/s$
2. Minimum speed $V_0$ to reach D is $\sqrt{105}m/s$
3. Normal force on the object by the track at C if it reaches D is 15 N
4. Normal force on the object by the track at B if it reaches D is 32.5 N

Q. A particle in a certain consevative force field has a potential energy given by $U=\frac{20xy}{z}$. The force exerted on it is

1. $\left(\frac{20y}{z}\right)\hat{i}+\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^2}\right)\hat{k}$
2. $-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^2}\right)\hat{k}$
3. $-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}-\left(\frac{20xy}{z^2}\right)\hat{k}$
4. $\left(\frac{20y}{z}\right)\hat{i}+\left(\frac{20x}{z}\right)\hat{j}-\left(\frac{20xy}{z^2}\right)\hat{k}$

Q. If the energy stored in spring '$\text{A}$' is $20\text{J}$, then energy stored in spring '$\text{B}$' is (under the same stretching force)

1. $10\text{J}$
2. $25\text{J}$
3. $40\text{J}$
4. $35\text{J}$

Q. Which of the following force is conservative force?

1. $\overrightarrow{F}=-3y\hat{i}-4x\hat{j}$
2. $\overrightarrow{F}=5y\hat{i}-5x\hat{j}$
3. $\overrightarrow{F}=3y\hat{i}+4x\hat{j}$
4. $\overrightarrow{F}=-5y\hat{i}-5x\hat{j}$

Q. For the path $PQR$ in a conservative force field (figure), the amount of work done in carrying a body from $P$ to $Q$ & from $Q$ to $R$ are $3\text{J}$ & $2\text{J}$ respectively. The work done in carrying the body from $P$ to $R$ will be

1. $7\text{J}$
2. $3\text{J}$
3. $5\text{J}$
4. zero

Q. The potential energy $U$ of a particle of mass $m=1\text{kg}$ moving in $x-y$ plane is given by $U=3x+4y,$ where $x$ and $y$ are in metre and $U$ is in joule. If initially particle was at rest, then its speed at $t=2\text{s}$ will be

1. $10\text{m/s}$
2. $6\text{m/s}$
3. $8\text{m/s}$
4. $7\text{m/s}$

Q. A non conservative force dissipates energy, while a conservative force does not dissipate energy

1. True
2. False

Q. When a non conservative force works on a system, total energy of the system remains constant.

1. False
2. True

Q. If $U_1,U_2$ represents "change in potential energy" of a body with mass $m$ moving from $A$ to $B$ along two different paths $1$, & $2$ (as shown in the figure) in the gravitational field. Then

1. $U_1=U_2$
2. $U_1>U_2$
3. $U_1<U_2$
4. $U_1\ge U_2$

Q. A particle, which is constrained to move along the $x-$ axis, is subjected to a force from the origin as $F\left(x\right)=-kx+a{x}^3$. Here $k$ and $a$ are positive constants. For $x=0$, the functional form of the potential energy $U\left(x\right)$ of particle is

1. 
2. 
3. 
4. 

Q.

A body is moving up an inclined plane of angle $\theta$ with an initial kinetic energy E. The coefficient of friction between the plane and the body is $\mu$. The work done against friction before the body comes to rest is

1. $\frac{\mu cos\theta }{Ecos\theta +sin\theta }$

2. $\mu Ecos\theta$

3. $\frac{\mu Ecos\theta }{\mu cos\theta -sin\theta }$

4. $\frac{\mu Ecos\theta }{\mu cos\theta +sin\theta }$

Q. Potential energy of a particle along $x-$ axis is given by $U=[-20+(x-2{)}^2]\text{J}$. Force on the particle at $x=0$ is

1. $6\text{N}$
2. $4\text{N}$
3. $8\text{N}$
4. $10\text{N}$

Q. Potential energy of a particle along $x-$ axis is given by $U=[-20+(x-2{)}^2]\text{J}$. Force on the particle at $x=0$ is

1. $6\text{N}$
2. $4\text{N}$
3. $8\text{N}$
4. $10\text{N}$

Q. Work done by a conservative force on a system is equal to

1. the change in kinetic energy of the system
2. the negative of the change in potential energy of the system
3. the change in total mechanical energy of the system
4. none of the above

Q. The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$, where $a$ and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is $D=[U_{x=\infty }-U_{\text{at equilibrium}}]$, $D$ is

1. $\frac{b^2}{2a}$
2. $\frac{b^2}{12a}$
3. $\frac{b^2}{4a}$
4. $\frac{b^2}{6a}$

Q. A ball of mass $400\text{g}$ is dropped to the ground from a height of $30\text{m}$. The ball bounces back multiple times before coming to rest. Which of the following statements is correct?

1. The potential energy of the ball increases as the ball bounces.
2. The work done by gravity increases if the ball bounces more number of times.
3. The work done by air resistance increases if the ball bounces more number of times.
4. The kinetic energy of the ball increases if the ball bounces more number of times.

Q.

If $W_1,W_{2}and{W}_3$ represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between $W_1,W_{2}and{W}_3$ [IIT JEE 2003]

1. $W_1>W_2>W_3$

2. $W_1=W_2=W_3$

3. $W_1<W_2<W_3$

4. $W_2>W_1>W_3$