Match the following? Column-I Column-II (a) F net (i) d K d t (b) F cons (ii) Δ U + Δ K (c) ∑ W non-cons (iii) - d U d x (d) Power (iv) d P d t

(a-iii), (b-ii), (c-ii), (d-ii)

A particle with total mechanical energy which is small and negative is under the influence of a one dimensional potential U x = x 4 4 - x 2 2 J , where x is in meters. At time t = 0 s , it is at x = - 0 . 5 m . Then at a later time it can be found,

between x = - 1 . 0 m to x = 0 . 0 m

between x = - 1 . 0 m to x = 1 . 0 m

between x = 0 . 0 m to x = 1 . 0 m

HARD

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A particle with an initial kinetic energy $E_{0}$ approaches the origin from $x = + \infty$ in the presence of a conservative potential. Consider following expressions for the corresponding potential energy $V (x)$ where $x$ is in meters:
(i) $V (x) = E_{0} [1 + \exp \{- {(x / x_{0})}^{2}\}]$
(ii) $V (x) = E_{0} [1 + 1.5 \exp \{- {((x - 5) / x_{0})}^{2}\}]$
(iii) $V (x) = E_{0} [1 - 0.5 \exp \{- {((x - 5) / x_{0})}^{2}\}]$
(iv) $V (x) = \frac{E_{0}}{{((x - 10) / x_{0})}^{2}} [1 - 0.5 \exp \{- {((x - 5) / x_{0})}^{2}\}]$
Here $x_{0} = 1 m .$ Select the correct statement(s):

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Important Questions on Work, Energy and Power

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

Which of the following is not a correct statement?

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

Two equal masses are attached to two each of a spring of spring constant

k

. The masses are pulled out symmetrically to stretch the spring by a length

x

over is natural length. The work done by the spring on each mass is

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

The potential energy of a particle in a central field has the form

U (r) = \frac{1}{r^{2}} - \frac{1}{r},

where

' r'

is the distance from the centre of the field. The magnitude of the maximum attractive force in Newton is

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A particle of mass

m

moves under the influence of the potential

V (x) = P / x^{2} - Q / x

. Here

P, Q

are real positive constants. The angular frequency of small oscillations of the particle around the equilibrium point is

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A particle is released from a height

H

. At a certain height its kinetic energy is half of its potential energy with reference to the surface of the earth. Height and speed of the particle at that instant are respectively

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

Which one of the following is not a conservative force?

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

The potential energy of a particle varies with distance

' x'

as shown in the graph. The force acting on the particle is zero at
Question Image

HARD

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

Consider two masses with

m_{1} > m_{2}

connected by a light inextensible string that passes over a pulley of radius

R

and moment of inertia

I

about its axis of rotation. The string does not slip on the pulley and the pulley turns without friction. The two masses are released from rest separated by a vertical distance

2 h .

When the two masses pass each other, the speed of the masses is proportional to

HARD

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A particle is moving in a circle of radius

r

under the action of a force

F = α r^{2}

which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy

= 0

for

r = 0

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

The graphs below show the magnitude of the force on a particle as it moves along the positive $X$ -axis from the origin to $X = X_{1}$ . The force is parallel to the $X$ -axis and conservative. The maximum magnitude $F_{1}$ has the same value for all graphs. Rank the situations according to the change in the potential energy associated with the force, least (or most negative) to greatest (or most positive).

Question Image

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

Potential energy as a function of

r

is given by

U = \frac{A}{r^{10}} - \frac{B}{r^{5}}

, where

r

is the interatomic distance,

A

and

B

are positive constants. The equilibrium distance between the two atoms will be :

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A block of mass

100 g

moving at a speed of

2 m s^{- 1}

compresses a spring through a distance

2 cm

before its speed is halved. Find the spring constant of the spring

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

The potential energy of an object is

U (x) = (5 x^{2} - 4 x^{3}) J

, where

x

is the position in meter. The position at which the force becomes zero is

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

Match the following?

Column-I		Column-II
(a)	$F_{net}$	(i)	$\frac{d K}{d t}$
(b)	$F_{cons}$	(ii)	$Δ U + Δ K$
(c)	$\sum W_{non-cons}$	(iii)	$\frac{- d U}{d x}$
(d)	$Power$	(iv)	$\frac{d P}{d t}$

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A rod of length

l

and mass

m

fixed at one end, is hanging vertically. The other end is now raised so that the rod makes an angle

30 °

with horizontal line. The work done in the process will be :

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

Hydrogen ion and singly ionized helium atom are accelerated, from rest, through the same potential difference. The ratio of final speeds of hydrogen and helium ions is close to:

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies

3.8 \times 10^{7}

J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take

g = 9.8 m s^{- 2}

EASY

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A particle of mass

m

moves in a circular orbit under the central potential field,

U (r) = \frac{- C}{r}

, where

C

is a positive constant. The correct radius - velocity graph of the particle's motion is :

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

The potential energy function for a two dimensional force is given by

U = 3 x^{3} y - 7 x

. The force that acts at the point

(x, y)

is (Take

\hat{i}

and

\hat{j}

as unit vectors along

X

- and

Y

- axes)

MEDIUM

Physics>Mechanics>Work, Energy and Power>Conservative Force and Potential Energy

A particle with total mechanical energy which is small and negative is under the influence of a one dimensional potential

U (x) = \frac{x^{4}}{4} - \frac{x^{2}}{2} J

, where

x

is in meters. At time

t = 0 s

, it is at

x = - 0.5 m

. Then at a later time it can be found,

Important Questions on Work, Energy and Power

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