Restoring Force

Q. The gravitational field in a region is given by the equation $\overrightarrow{E}=(5\hat{i}+12\hat{j})\text{N/kg}$. If a particle of mass $2\text{kg}$ is moved from the origin to the point $(12\text{m},5\text{m})$ in this region, the change in the gravitational potential energy is

1. $-480\text{J}$
2. $480\text{J}$
3. $-240\text{J}$
4. $240\text{J}$

Q. The variation of potential energy $\left(U\right)$ with position $\left(x\right)$ of a system is shown in the figure. The conservative force acting on the system is best represented by

1. 
2. 
3. 
4. 

Q. A body of mass $2\text{kg}$ is projected from origin with a velocity $3\text{m/s}$ along positive $x-$ axis on which a force $-x^2$ is acting. Find the maximum distance from origin upto which the body can go. ($x$ is the position of particle)

1. $6\text{m}$
2. $5\text{m}$
3. $3\text{m}$
4. $4\text{m}$

Q. A particle of mass ${10}^{-2}\text{kg}$ is moving along the positive $x$-axis under the influence of a force $F\left(x\right)=-\frac{\alpha }{8x^2}$, where $\alpha ={10}^{-2}{\text{N.m}}^2$. At time $t=0$, it is at $x=1.0\text{m}$ and velocity is $v=0$. Find the velocity when it reaches $x=0.5\text{m}$. Consider the magnitude along with sign to represent the velocity.

1. $-0.5\text{m/s}$
2. $-1\text{m/s}$
3. $1\text{m/s}$
4. $2.5\text{m/s}$

Q. A particle of mass ${10}^{-2}\text{kg}$ is moving along the positive $\text{x-axis}$ under the influence of a force $F\left(x\right)=-\frac{K}{2x^2}$ where $K={10}^{-2}{\text{Nm}}^2$. At time $t=0$, particle is at $x=1\text{m}$ and its velocity is $v=0$. The time at which it reaches $x=0.25\text{m}$ is

1. $1.7\text{s}$
2. $1.2\text{s}$
3. $1.9\text{s}$
4. $1.5\text{s}$

Q. A projectile starts from the origin at time $t=0$ and moves in the $\text{xy-plane}$ with a constant acceleration $\alpha$ in the $\text{y-direction}$ and constant speed along $x$ direction. If equation of motion is $y=\beta x^2$ then its velocity component in the $\text{x-direction}$ will be

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

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

1. 
2. 
3. 
4. 

Q. A particle of mass ${10}^{-2}\text{kg}$ is moving along the positive $x$-axis under the influence of a force $F\left(x\right)=-\frac{\alpha }{8x^2}$, where $\alpha ={10}^{-2}{\text{N.m}}^2$. At time $t=0$, it is at $x=1.0\text{m}$ and velocity is $v=0$. Find the velocity when it reaches $x=0.5\text{m}$. Consider the magnitude along with the sign to represent the velocity.

1. $-0.5\text{m/s}$
2. $-1\text{m/s}$
3. $1\text{m/s}$
4. $2.5\text{m/s}$

Q. At time $t=0$ a particle starts moving along the x axis. If its kinetic energy increases uniformly with $t$, the net force acting on it must be

1. Constant
2. Proportional to $t$
3. Inversely proportional to $t^2$
4. Proportional to $1/\sqrt{t}$

Q.

The potential energy for a force field $\overline{F}$ is given by $\cup$(x, y) = cos(x+y). The force acting on a particle at position given by coordinates (0, $\pi$/4) is

1. 

2. 

3. 

4. 

Q.

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

1. 

2. 

3. 

4. 

Q. A particle of mass ${10}^{-2}\text{kg}$ is moving along the positive $x$-axis under the influence of a force $F\left(x\right)=-\frac{\alpha }{8x^2}$, where $\alpha ={10}^{-2}{\text{N.m}}^2$. At time $t=0$, it is at $x=1.0\text{m}$ and velocity is $v=0$. Find the velocity when it reaches $x=0.5\text{m}$. Consider the magnitude along with the sign to represent the velocity.

1. $-0.5\text{m/s}$
2. $-1\text{m/s}$
3. $1\text{m/s}$
4. $2.5\text{m/s}$

Q.

A particle which is constrained to move along 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 constants. For $x\ge 0$, the functional from of the potential energy $U_{\left(x\right)}$of the particle is [IIT JEE 2002]

1. 

2. 

3. 

4. 

Q. The potential energy of a point particle is given by the expression $V\left(x\right)=-\alpha x+\beta \sin \left(x/\gamma \right)$. A dimensionless combination of the constants $\alpha ,\beta$ and $\gamma$ is

1. $\alpha /\beta \gamma$
2. $\gamma /\alpha \beta$
3. $\alpha \gamma /\beta$
4. ${\alpha }^2/\beta \gamma$

Q. A projectile is given an initial velocity of $(\hat{i}+2\hat{j})\text{m/s}$, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g=10{\text{m/s}}^2$, the equation of its trajectory is :

1. $y=x-5x^2$
2. $4y=2x-5x^2$
3. $4y=x-5x^2$
4. $y=2x-5x^2$

Q. For the arrangement of forces in Problem $81,$ a $2.00kg$ particle is released at $x=5.00m$ with an initial velocity of $3.45m/s$ in the negative direction of the x axis. (a) If the particle can reach $x=0m,$ what is its speed there, and if it cannot, what is its turning point? Suppose, instead, the particle is headed in the positive x direction when it is released at $x=5.00m$ at speed $3.45m/s.$ (b) If the particle can reach $x=13.0m,$ what is its speed there, and if it cannot, what is its turning point?

Q. The potential energy for a force field $\overline{F}$ is given by $U(x,y)=sin(x+y)$. The force acting on the particle mass $m$at $(0,\frac{\pi }{4})$ is

1. $1$
2. $\sqrt{2}$
3. $\frac{1}{\sqrt{2}}$
4. $0$

Q. A $3.0kg$ toy car moves along an $x$ axis with a velocity given by $\overrightarrow{V}=-2.0t^3\overrightarrow{i}m/s$, with $t$ in seconds. For $t>0$, what are the torque $\overrightarrow{r}$ on the car, both calculated about the origin?

Q. The potential energy of a particle varies with distance x as shown in the graph. The force acting on the particle is zero at:

1. C
2. D
3. B and C
4. A and D

Q. Calculate the force F(y) associated the following one-dimensional potential energies:

Q.

A particle which is constrained to move along 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 constants. For $x\ge 0$, the functional from of the potential energy $U_{\left(x\right)}$of the particle is [IIT JEE 2002]

1. 

2. 

3. 

4. 

Q. In Fig 13-37 a, particle A is fixed in placed at x= -0.20 m on the x aixs and particle B with a mass 1.0 kg is fixed in place at the origin particle C are ( not shown ) can be moved along the x axis , between particle B and x= $\infty$ Figure 13-37b shows the x component $F_{netx}$ of the net gravitational force on particle B due to practices A and C, as a functional of position x particle c. The plot actually extend to the right approaching an asymptote of $-4.17\times {10}^{-10}$ N as $x\to \infty$ what are the masses of (a) particles A and (b) particles (C)

Q. If a particle moves alone a straight line whose equation is $y=mx$ then what will be its angular momentum about its origin? Explain your answer.

Q. Potential energy v/s displacement curve for one dimensional field is shown. Force at A and B is respectively.

1. Positive, Positive
2. Positive, Negative
3. Negative, Positive
4. Negative, Negative