The potential energy \(\mathrm{U}\) of a system is given by $\mathrm{U}=$ $\mathrm{A}$ $-$ ${\mathrm{Bx}}^2$ (where \(\mathrm{x}\) is the position of its particle and \(\mathrm{A},\) \(\mathrm{B}\) are constants). The magnitude of the force acting on the particle is:
1. constant

2. proportional to \(\mathrm{x}\)

3. proportional to ${\mathrm{x}}^2$

4. proportional to $\left(\frac{1}{\mathrm{x}}\right)$

A person-1 stands on an elevator moving with an initial velocity of 'v' & upward acceleration 'a'. Another person-2 of the same mass m as person-1 is standing on the same elevator. The work done by the lift on the person-1 as observed by person-2 in time 't' is:

1.  $\mathrm{m}\left(\mathrm{g}\right)$ $+$ $\mathrm{a}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

2.  $-\mathrm{mg}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

3.  0

4.  $\mathrm{ma}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

The figure shows the potential energy function U(x) for a system in which a particle is in a one-dimensional motion. What is the direction of the force when the particle is in region AB? (symbols have their usual meanings)

1.  The positive direction of x

2.  The negative direction of X

3.  Force is zero, so direction not defined

4.  The negative direction of y

A block of mass 'm' is connected to a spring of force constant K. Initially, the block is at rest and the spring is relaxed. A constant force F is applied horizontally towards the right. The maximum speed of the block will be:

A cord is used to vertically lower a block of mass m by a distance d at a constant downward acceleration of $\frac{\text{'}\mathrm{g}\text{'}}{4}$. The work done by the chord on the block will be:

1.  $\frac{3}{4}$mgd

2.  -$\frac{3}{4}$mgd

3.  $\frac{1}{4}$mgd

4.   -$\frac{1}{4}$mgd

An engine pumps liquid of density d continuously through a pipe of cross-sectional area A. If the speed with which liquid passes through the pipe is v, then the rate at which kinetic energy is being imparted to the liquid by the pump is: 

The work done in increasing the length of a massless spring from its natural length 15 cm to 15.1 cm is 20 J. Work done in increasing the length from 15.1 cm to 15.2 cm will be:

A body of mass m moving at a certain speed suffers a perfectly inelastic collision with a body of mass M at rest. The ratio of the final kinetic energy of the system to the initial kinetic energy will be:

1. \(m \over {m + M}\)

2. \(M \over {m + M}\)

3. \({m + M} \over m\)

4. \({m + M} \over M\)

An automobile of mass m accelerates from rest, while the engine supplies constant power P. The speed of the automobile as a function of time is given by:

1.  $\mathrm{v} = {\left(\frac{2\mathrm{Pt}}{\mathrm{m}}\right)}^{1/2}$

2.  $\mathrm{v} = {\left(2\mathrm{Ptm}\right)}^{1/3}$

3.  $\mathrm{v} = {\left(\frac{8}{9}\frac{\mathrm{P}}{\mathrm{m}}{\mathrm{t}}^3\right)}^{1/2}$

4.  $\mathrm{v} = {\left(2{\mathrm{Pt}}^3\mathrm{m}\right)}^{1/2}$

The potential energy of a particle of mass 1 kg free to move along the X-axis is given by \(U(x) = (3x^2-4x+6)~\text{J}\). The force acting on the particle at x = 0 will be:

1.  2$\hat{\mathrm{i}}$ N

2.  -4$\hat{\mathrm{i}}$ N

3.  5$\hat{\mathrm{i}}$ N

4.  4$\hat{\mathrm{i}}$ N