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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When a ceiling fan is switched off, its angular velocity fall to half while it makes 36 rotations. How many more rotations will it make before coming to rest? (Assume uniform angular retardation)

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Important Questions on Motion of System of Particles and Rigid Body

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

Four particles, each of mass

M

and equidistant from each other, move along a circle of radius

R

under the action of their mutual gravitational attraction. The speed of each particle is

MEDIUM

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

A rigid massless rod of length $3 l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be

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HARD

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

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

EASY

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

Suppose that the angular velocity of rotation of the Earth is increased. Then, as a consequence,

HARD

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

A straight rod of length

L

extends from

x = a

x = L + a .

The gravitational force it exerts on a point mass 'm' at

x = 0,

if the mass per unit length of the rod is

A + B x^{2},

is given by:

EASY

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

A star mass $M$ (equal to the solar mass) with a planet (much smaller than the star) revolves around the star in a circular orbit. The velocity of the star with respect to the centre of mass of star-planet system is shown below:

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The radius of the planet's orbit is closest to ( $1 AU = Distance Earth and Sun$ )

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

A very long (length $L$ ) cylindrical galaxy is made of uniformly distributed mass and has radius $R (R < < L)$ . A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of the star is $T$ and its distance from the galaxy's axis is $r$ , then

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

The depth

d

at which the value of acceleration due to gravity becomes

\frac{1}{n}

times the value at the earth's surface is

(R

= radius of the earth

)

HARD

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

An automobile moves on a road with a speed of

54 k m h^{- 1}

. The radius of its wheels is

0.45 m

and the moment of inertia of the wheel about its axis of rotation is

3 k g m^{2}

. If the vehicle is brought to rest in

15 s

, the magnitude of average torque transmitted by its brakes to the wheel is:

EASY

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

Two stones of masses $m$ and $2 m$ are whirled in horizontal circles, the heavier one in radius $\frac{r}{2}$ and the lighter one in radius $r$ . The tangential speed of lighter stone is $n$ times that of the tangential speed of the heavier stone. They are reported to experience the same centripetal force. The value of $n$ is,

EASY

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

Starting from the center of the earth having radius,

R

, the variation of

g

(acceleration due to gravity) is shown by

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

A wheel of radius "rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road a small bob of the mud separates from wheel at the highest point and touches the point $B$ on the road as shown in the figure. Then $A B$ is (g-acceleration due to gravity)

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

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

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

Two satellites

S_{1}

and

S_{2}

are revolving around a planet in the opposite sense in coplanar circular concentric orbits. At time

t = 0,

the satellites are farthest apart. The periods of revolution of

S_{1}

and

S_{2}

are

3 h

and

24 h

respectively. The radius of the orbit of

S_{1}

3 \times 10^{4} km .

Then the orbital speed of

S_{2}

as observed from

EASY

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

The variation of acceleration due to gravity

g

with distance

d

from the centre of the earth is best represented by (

R

= Earth's radius):

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

A ball is launched from the top of Mount Everest which is at an elevation of

9000 m

. The ball moves in a circular orbit around the earth. The acceleration due to gravity near the earth's surface is

g

. The magnitude of the ball's acceleration while in orbit is,

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

The International space station is maintained in a nearly circular orbit with a mean altitude of

330 km

and a maximum of

410 km

. An astronaut is floating in the space station's cabin. The acceleration of astronauts as measured from the earth is-

EASY

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

If the change in the value of

g

at a height

h

above the surface of the earth is the same as at a depth

x

below it, then (both

x

and

h

being much smaller than the radius of the earth)

EASY

Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

At what height from the surface of earth the gravitation potential and the value of g are

- 5.4 \times 10^{7} J k g^{- 2}

and

6.0 m s^{- 2}

respectively? Take the radius of earth as

6400 km

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Physics>Mechanics>Motion of System of Particles and Rigid Body>Translation and Rotation of a Body

Consider two uniform discs of the same thickness and different radii

R_{1} = R

and

R_{2} = α R

made of the same material. If the ratio of their moments of inertia

I_{1}

and

I_{2}

, respectively, about their axes is

I_{1} : I_{2} = 1 : 16

then the value of

α

is :

When a ceiling fan is switched off, its angular velocity fall to half while it makes 36 rotations. How many more rotations will it make before coming to rest? (Assume uniform angular retardation)

Important Questions on Motion of System of Particles and Rigid Body

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