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

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A block of mass $2 kg$ is kept at origin at $t = 0$ and is having velocity $4 \sqrt{5} m / s$ in positive $x -$ direction. The only force on it is a conservative and its potential energy is defined as $U = - x^{3} + 6 x^{2} + 15$ ( $SI$ units). Its velocity when the force acting on it is minimum (after the time $t = 0$ ) is

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Important Questions on Work, Energy and Power

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A uniform chain of mass

M

and length

L

is lying on a smooth horizontal table, with half of its length hanging down. The work done in pulling the entire chain up the table is

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Science>Physics>Work, Energy and Power>Potential Energy

A mass of $0.5 kg$ moving with a speed of $1.5 m s^{- 1}$ on a horizontal smooth surface, collides with a nearly weightless spring of force constant $k = 50 N m^{- 1}$ . The maximum compression of the spring would be

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A body is moving unidirectionally under the influence of a source of constant power. The square of its displacement in time

t

is proportional to

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Science>Physics>Work, Energy and Power>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)

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Science>Physics>Work, Energy and Power>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}

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Science>Physics>Work, Energy and Power>Potential Energy

If a spring of spring constant

200 N m^{- 1}

is compressed by

5 cm

, then the energy stored in the spring is

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An asteroid of mass

m

is approaching earth, initially at a distance

10 R_{e}

with speed

v_{i} .

It hits Earth with a speed

v_{f},

(R_{e}

and

M_{e}

are radius and mass of earth

)

, then

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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).

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A particle

(m = 1 k g)

slides down a frictionless track

(A O C)

starting from rest at a point

A

(height

2 m

). After reaching

C,

the particle continues to move freely in air as a projectile. When it reaches its highest point

P

(height

1 m

), the kinetic energy of the particle (in

J

) is: (Figure drawn is schematic and not to scale; take

g = 10 m s^{- 2}

) ______________.
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Science>Physics>Work, Energy and Power>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:

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Science>Physics>Work, Energy and Power>Potential Energy

A mass of

2 kg

, initially at a height of

1.2 m

above an uncompressed spring with spring constant

2 \times 10^{4} N m^{- 1},

is released from rest to fall on the spring. Taking the acceleration due to gravity as

10 m s^{- 2}

and neglecting the air resistance, the compression of the spring in

mm

HARD

Science>Physics>Work, Energy and Power>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

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Science>Physics>Work, Energy and Power>Potential Energy

A body is released from a height of

30 m

vertically downwards. The speed of the body at which potential energy is twice that of kinetic energy is (Acceleration due to gravity

= 10 m s^{- 2}

)

EASY

Science>Physics>Work, Energy and Power>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

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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

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Science>Physics>Work, Energy and Power>Potential Energy

A large block of wood of mass $M = 5.99 kg$ is hanging from two long massless cords. A bullet of mass $m = 10 g$ is fired into the block and gets embedded in it. The (block $+$ bullet) then swing upwards, their center of mass rising a vertical distance $h = 9.8 cm$ before the (block $+$ bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before the collision is: (Take $g = 9.8 {m s}^{- 2}$ )

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HARD

Science>Physics>Work, Energy and Power>Potential Energy

The potential energy

(U)

of a diatomic molecule is a function dependent on

r

(interatomic distance) as

U = \frac{α}{r^{10}} - \frac{β}{r^{5}} - 3

where,

α

and

β

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

{(\frac{2 α}{β})}^{\frac{1}{a}},

where

a =

______ .

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If the potential energy between two molecules is given by

U = \frac{A}{r^{6}} - \frac{B}{r^{12}},

then at equilibrium, separation between molecules, and the potential energy are:

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Science>Physics>Work, Energy and Power>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,

HARD

Science>Physics>Work, Energy and Power>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

A block of mass 2 kg is kept at origin at t=0 and is having velocity 45 m/s in positive x-direction. The only force on it is a conservative and its potential energy is defined as U=-x3+6x2+15 ( SI units). Its velocity when the force acting on it is minimum (after the time t=0 ) is

Important Questions on Work, Energy and Power

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