Conservative and Non Conservative Forces

Q.

Give three examples when work done is zero.

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. The particle of mass $m$ is moving in time $t$ on a trajectory given by, $\overrightarrow{r}=10\alpha t^2\hat{i}+5\beta (t-5)\hat{j}$
Where $\alpha$ and $\beta$ are dimensional constants.
The angular momentum of the particle becomes the same, as it was for $t=0$, at time $t=$ seconds.

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.

A particle free to move along the x-axis has potential energy given by $U\left(x\right)=k[1-exp(-x{)}^2]$ for $-\infty \le x\le +\infty$, where k is a positive constant of appropriate dimensions. Then

1. If its total mechanical energy is k/2, it has its minimum kinetic energy at the origin

2. For any finite non-zero value of x, there is a force directed away from the origin

3. For small displacements from x = 0, the motion is simple harmonic

4. At point away from the origin, the particle is in unstable equilibrium

Q. The potential energy of a system is represented in the first figure. The force acting on the system will be represented by

1. 
2. 
3. 
4. 

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.

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.

Work done in the motion of a body over a closed loop is zero for every force in nature.

1. True
2. False

Q. The masses and radii of earth and moon are $M_1,R_1\text{and}{M}_2,R_2$ respectively. Their centers are distance $d$ apart. The minimum velocity with which a particle of mass $m$ should be projected from a point midway between their centre so that it escapes to infinity is:-

1. $2\sqrt{\frac{G}{d}(M_1+M_2)}$
2. $2\sqrt{\frac{2G}{d}(M_1+M_2)}$
3. $2\sqrt{\frac{Gm}{d}(M_1+M_2)}$
4. $2\sqrt{\frac{Gm(M_1+M_2)}{d(R_1+R_2)}}$

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.

Work done by conservative forces is equal to increase in potential energy.

1. True
2. False

Q. Is the work energy theorem for variable force is applicable for both conservative and non-conservative forces or not?

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.

A particle of mass m moves with constant speed along a circular path of radius r under the action of a force F. Its speed is:

1. $\sqrt{\frac{rF}{m}}$

2. $\sqrt{\frac{F}{r}}$

3. $\sqrt{Fmr}$

4. $\sqrt{\frac{F}{mr}}$

Q. Which of the following statements regarding conservative forces is NOT correct?

1. Total energy remains constant
2. Work done in round trip is zero
3. Work done is dependent on path
4. Work done is completely recoverable

Q. In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \ne B$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned; $\overrightarrow{p}$ is the linear momentum $\overrightarrow{L}$ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for the path.

List - I	List - II
P. $\overrightarrow{r}\left(t\right)=\alpha t\hat{i}+\beta t\hat{j}$	1. $\overrightarrow{p}$
Q. $\overrightarrow{r}\left(t\right)=\alpha cos\left(\omega t\right)\hat{i}+\beta sin\left(\omega t\right)\hat{j}$	2. $\overrightarrow{L}$
R. $\overrightarrow{r}\left(t\right)=\alpha \left(cos\right(\omega t)\hat{i}+sin(\omega t\left)\hat{j}\right)$	3. K
S. $\overrightarrow{r}\left(t\right)=\alpha t\hat{i}+\frac{\beta }{2}{t}^2\hat{j}$	4. U
	5. E

1. $P\to 1,2,3,4,5;Q\to 2,5;$ \ R \rightarrow 2, 3, 4, 5 ; \ S \rightarrow 5$
2. $P\to 1,2,3,4,5;Q\to 3,5;$ \ R \rightarrow 2, 3, 4, 5 ; \ S \rightarrow 2, $
3. $P\to 2,3,4;Q\to 5;$\ R \rightarrow 1, 2, 4 ; \ S \rightarrow 2, 5$
4. $P\to 1,2,3,5;Q\to 2,5;$ \ R \rightarrow 2, 3, 4, 5 ; \ S \rightarrow 2, 5$

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, 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.

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

1. The change in kinetic energy of the system

2. The change in potential energy of the system

3. The change in total mechanical energy of the system

4. None of the above

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

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

Q. A spring is initially compressed by $2\text{m}$. Find out the work done required to compress another $6\text{m}$ if spring constant $k=2\text{N/m}$.

1. $60\text{J}$
2. $32\text{J}$
3. $68\text{J}$
4. $36\text{J}$

Q. Friction is a conservative force.

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. The potential energy of a conservative system is given by $U=a{x}^2-bx$
where a and b are positive constants. Find the equilibrium position and discuss whether the equilibrium is stable, unstable or neutral.

1. $x=-\frac{b}{2a}$ is the stable equilibrium position.
2. $x=0$ is the stable equilibrium position
3. $x=\frac{b}{2a}$ is a unstable equilibrium position
4. $x=\frac{b}{2a}$ is the stable equilibrium position

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

1. False
2. True

Q. A particle moves along a curve of unknown shape but magnitude of force $\overrightarrow{F}$ is constant and always acts along the tangent to the curve. Then,

1. $\overrightarrow{F}$ must be conservative
2. $\overrightarrow{F}$ may be conservative
3. $\overrightarrow{F}$ must be non-conservative
4. $\overrightarrow{F}$ may be non-conservative

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. 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