A particle of mass m is moving along the side of a square of side 'a', with a uniform speed υ in the x-y plane as shown in the figure: Which of the following statements is false for the angular momentum L → about the origin?

L → = m υ 2 R k ^ when the particle is moving from D to A.

L → = m υ R 2 + a k ^ when the particle is moving from B to C.

L → = - m υ 2 R k ^ when the particle is moving from A to B.

L → = m υ R 2 + a k ^ when the particle is moving from C to D.

HARD

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A disc is freely rotating with an angular speed $ω$ on a smooth horizontal plane. Suddenly it is hooked at a rigid peg $P$ and then the disc rotates without bouncing about the peg $P$ . Find the angular velocity of the disc after the impact. [Length of the peg from the disc is neglected]

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Important Questions on Systems of Particles and Rotational Motion

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Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A hoop of radius

r

and mass

m

rotating with an angular velocity

ω_{0}

is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A particle of mass m is moving along the side of a square of side 'a', with a uniform speed

υ

in the x-y plane as shown in the figure:

Question Image

Which of the following statements is false for the angular momentum

\vec{L}

about the origin?

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

The time dependence of the position of a particle of mass

m = 2

is given by

\vec{r} (t) = 2 t \hat{i} - 3 t^{2} \hat{j}

. Its angular momentum, with respect to the origin, at time

t = 2

is:

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A thin smooth rod of length

L

and mass

M

is rotating freely with angular speed

ω_{0}

about an axis perpendicular to the rod and passing through center. Two beads of mass

m

and negligible size are at the center of the rod initially. The beads of mass

m

and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be:

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A disc of the moment of inertia

I_{1}

is rotating in a horizontal plane about an axis passing through a centre and perpendicular to its plane with constant angular speed

ω_{1}

Another disc of the moment of inertia

I_{2}

having zero angular speed is placed coaxially on a rotating disc. Now both the disc are rotating with the constant angular speed

ω_{2}

. The energy lost by the initial rotating disc is

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

Two coaxial discs, having moments of inertia

I_{1}

and

\frac{I_{1}}{2}

, are rotating with respective angular velocities

ω_{1}

and

\frac{ω_{1}}{2}

(in the same direction), about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If

E_{f}

and

E_{i}

are the final and initial total energies, then

(E_{f} - E_{i})

is:

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A cubical block of side $30 cm$ is moving with velocity $2 m s^{- 1}$ on a smooth horizontal surface. The surface has a bump at a point $O$ as shown in the figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is :

Question Image

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A solid sphere of radius

r

is revolving about one of its diameters with an angular velocity

ω .

If it suddenly expands uniformly so that its radius increases to

n

times its original value, then its angular velocity becomes

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A thin circular ring of mass

m

and radius

R

is rotating about its axis perpendicular to the plane of the ring with a constant angular velocity

ω

. Two point particles each of mass

M

are attached gently to the opposite ends of a diameter of the ring. The ring now rotates, with an angular velocity

\frac{ω}{2}

. Then, the ratio

\frac{m}{M}

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A horizontal disk of moment of inertia

4.25 kg - m^{2}

with respect to its axis of symmetry is spinning counter clockwise at

15

revolutions per second about its axis, as viewed from above. A second disk of moment of inertia

1.80 kg - m^{2}

with respect to its axis of symmetry is spinning clockwise at

25

revolutions per second as viewed from above about the same axis and is dropped on top of the first disk. The two disks stick together and rotate as one about their axis of symmetry. The new angular velocity of the system as viewed from above is close to.

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

Three point masses each of mass

m

are kept at the corners of an equilateral triangle of side

L

. The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to

(\cos 30 ° = s i n 60 ° = \frac{\sqrt{3}}{2})

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A bob of mass m attached to an inextensible string of length

l

is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed

ω

rad/s about the vertical. About the point of suspension :

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A uniformly thick wheel with moment of inertia

I

and radius

R

is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses

m_{1}

and

m_{2} (m_{1} > m_{2})

are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when

m_{1}

descends by a distance

h

is:

HARD

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A particle of mass

20 g

is released with an initial velocity

5 m s^{- 1}

along the curve from the point

A,

as shown in the figure. The point

A

is at height

h

from point

B .

The particle slides along the frictionless surface. When the particle reaches point

B,

its angular momentum about

O

will be: (Take

g = 10 m s^{- 2}

)

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A particle is moving uniformly along a straight line as shown in the figure. During the motion of the particle from $A$ to $B$ , the angular momentum of the particle about ${}^{'}O^{'}$

Question Image

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius

R_{e} .

By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become

\sqrt{\frac{3}{2}}

times larger. Due to this the farthest distance from the centre of the earth that the satellite reaches is

R

. Value of

R

is :

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are

0.1 kg - m^{2}

and

10 rad s^{- 1}

respectively while those for the second one are

0.2 kg - m^{2}

and

5 rad s^{- 1}

respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is :

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A ball of mass

160 g

is thrown up at an angle of

60 °

to the horizontal at a speed of

10 m s^{- 1}

. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly

(g = 10 m s^{- 2})

EASY

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

Two rotating bodies,

A

and

B

of masses,

m

and

2 m

with moments of inertia

I_{A}

and

I_{B} (I_{B} > I_{A})

have equal kinetic energy of rotation. If,

L_{A}

and

L_{B}

be their angular momenta, respectively, then,

MEDIUM

Physics>Mechanics>Systems of Particles and Rotational Motion>Torque

A particle of mass $2 kg$ is on a smooth horizontal table and moves in a circular path of radius $0.6 m$ . The height of the table from the ground is $0.8 m$ . If the angular speed of the particle is $12 rad s^{- 1}$ , the magnitude of its angular momentum about a point on the ground right under the center of the circle is:

A disc is freely rotating with an angular speed ω on a smooth horizontal plane. Suddenly it is hooked at a rigid peg P and then the disc rotates without bouncing about the peg P. Find the angular velocity of the disc after the impact. [Length of the peg from the disc is neglected]

Important Questions on Systems of Particles and Rotational Motion

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A disc is freely rotating with an angular speed $ω$ on a smooth horizontal plane. Suddenly it is hooked at a rigid peg $P$ and then the disc rotates without bouncing about the peg $P$ . Find the angular velocity of the disc after the impact. [Length of the peg from the disc is neglected]