Test: MCQs (One or More Correct Option): Rotational Motion | JEE Advanced


21 Questions MCQ Test Physics 35 Years JEE Main & Advanced Past year Papers | Test: MCQs (One or More Correct Option): Rotational Motion | JEE Advanced


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Attempt Test: MCQs (One or More Correct Option): Rotational Motion | JEE Advanced | 21 questions in 55 minutes | Mock test for JEE preparation | Free important questions MCQ to study Physics 35 Years JEE Main & Advanced Past year Papers for JEE Exam | Download free PDF with solutions
*Multiple options can be correct
QUESTION: 1

Two particles A and B initially at rest, move towards each other under mutual force of attraction. At  the instant when the speed of A  is V and the speed of B is 2V, the speed of the centre of mass of the system is

Solution:

We know that Fext = Mac.m. ... (i)

We consider the two particles in a system. Mutual force of attraction is a internal force. There are no external forces acting on the system. From (i)

ac.m. = 0 ⇒ vc.m. = constant.
Since, initially the vc.m. = 0

∴ Finally vc.m. = 0

*Multiple options can be correct
QUESTION: 2

A  mass M moving with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin

Solution:

Angular momentum

L = Momentum × perpendicular distance of line of action of momentum w.r.t point of rotation

L = Mv × y.

The quantities on the right side of the equation are not changing.
The magnitude is constant. The direction is also constant.

*Multiple options can be correct
QUESTION: 3

When a bicycle is in motion, the force of friction exerted by the ground on the two wheels is such that it acts

Solution:

When the cycle is not pedalled but the cycle is in motion (due to previous effort) the wheels move in the direction such that the centre of mass of the wheel move forward. Rolling friction will act in the opposite direction to the relative motion of the centre of mass of the body with respect to ground. Therefore the rolling friction will act in backward direction in both the wheels. The sliding friction will act in the forward direction of rear wheel during pedalling.

*Multiple options can be correct
QUESTION: 4

A particle of mass m is projected with a velocity v making an angle of 45° with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection when the particle is at its maximum height h is

Solution:

Angular momentum = (momentum) × (perpendicular distance of the line of action of momentum from the axis of rotation) Angular momentum about O

From (i) and (ii)

Also, from (i) and (ii)  

*Multiple options can be correct
QUESTION: 5

A uniform bar of length 6a and mass 8m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speed 2v and v, respectively, strike the bar [as shown in the fig.] and stick to the bar after collision. Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by w, E and vc respectively, we have after collision

Solution:

Applying conservation of linear momentum 2m (– v) + m (2v) + 8m × 0 = (2m + m + 8m) vc

⇒ vc = 0

Applying conservation of angular momentum about centre of mass

I = 30ma2 ...(ii)

From (i) and (ii)

Energy after collision, 

 

*Multiple options can be correct
QUESTION: 6

The moment of inertia of a thin square plate ABCD, Fig., of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is

wh ere I1, I 2 , I3 andI4 are respectively the momen ts of intertial about axis 1, 2, 3 and 4 which are in the plane of the plate.

Solution:

To find the moment of inertia of ABCD about an axis passing through the centre O and perpendicular to the plane of the plate, we use perpendicular axis theorem. If we consider ABCD to be in the X-Y plane then we know that

Also, Izz' = I3 + I4

Adding (i) and (ii),

But I1 = I2 and I3 = I4

(By symmetry)

 

*Multiple options can be correct
QUESTION: 7

A tube of length L is filled completely with an incomressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity ω. The force exerted by the liquid at the other end is

Solution:

The force acting on the mass of liquid dm of length dx at a distance x from the axis of rotation O.

dF = (dm) x ω2

where  M/L is mass of liquid in unit length.

∴The force acting at the other end is for the whole liquid in tube.

 

*Multiple options can be correct
QUESTION: 8

A car is moving in a circular horizontal track of radius 10 m with a constant speed of 10 m/s. A pendulum bob is suspended from the roof of the car by a light rigid rod of length 1.00 m. The angle made by the rod with track is

Solution:

When the car is moving in a circular horizontal track of radius 10 m with a constant speed, then the bob is also undergoing a circular motion. The bob is under the influence of two forces.

(i) T (tension in the rod)

(ii) mg (weight of the bob)

Resolving tension, we get

Tcosθ = mg          ... (i)  

And Tsinθ = mv2/r      ... (ii)

(Here T sin θ is producing the necessary centripetal force for circular motion)

Dividing (ii) by (i), we get

 

*Multiple options can be correct
QUESTION: 9

Let I be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle q with AB. The moment of inertia of the plate about the axis CD is then equal to

Solution:

A'B' ⊥ AB and C' D'⊥ CD

From symmetry IAB = IA'B' and ICD = IC'D'
From theorem of

perpendicular axes,

Izz = IAB + IA'B' = ICD + IC'D'

⇒ 2IAB = 2ICD

 IAB = ICD

*Multiple options can be correct
QUESTION: 10

The torqueτ on a body about a given point is found to be equal to A × L where A is a constant vector, and L is the angular momentum of the body about that point. From this it follows that

Solution:

KEY CONCEPT 

Given that

From cross-product rule, uur  is always perpendicular to the

By the dot product definition

Differentiating with respect to time

Since,

⇒ L = constant

Thus, the magnitude of L always remains constant.

 

 

*Multiple options can be correct
QUESTION: 11

A solid cylinder is rolling down a rough inclined plane of inclination θ. Then

Solution:

As shown in the figure, the component of weight mg sinθ tends to slide the point of contact (of the cylinder with inclined plane) along its direction. The sliding friction acts in the opposite direction to oppose this relative motion.
Because of frictional force the cylinder rolls.

Thus frictional force adds rotation but hinders translational motion.
Applying Fnet = ma along the direction of inclined plane,

we get  mg sin 

where ac = acceleration of centre of mass of the cylinder

From (i) and (ii)

If θ is reduced, frictional force is reduced.

*Multiple options can be correct
QUESTION: 12

If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that   

Solution:

Due to internal forces acting in the system, the kinetic and potential energy may change with time.
Also zero external force may create a torque if the line of action of forces are along different direction. Thus the torque will change the angular momentum of the system.

*Multiple options can be correct
QUESTION: 13

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then,

Solution:

 is the velocity of centre of the sphere, then

(b) is the correct option.

 

(c) is the correct option.

*Multiple options can be correct
QUESTION: 14

Two solid spheres A and B of equal volumes but of different densities dAand dBare connected by a string. They are fully immersed in a fluid of density dF. They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

Solution:

Let V be the volume of spheres.
For equilibrium of A :

T + VdAg = Vdfg

(a) is the correct option

For equilibrium of B :

 

(b) is the correct option

(d) is the correct option.

*Multiple options can be correct
QUESTION: 15

A thin ring of mass 2 kg and radius 0.5 m is rolling without on a horizontal plane with velocity 1 m/s. A small ball of mass 0.1 kg, moving with velocity 20 m/s in the opposite direction hits the ring at a height of 0.75 m and goes vertically up with velocity 10 m/ s. Immediately after the collision

Solution:

The frictional force between   

the ring and the ball is impulsive. The angular impulse created by this force tends to decrease the angular speed of the ring about O.
After the collision the angular speed decreases but the ring remains rotating in the anticlock wise direction.
Therefore the friction between the ring and the ground (at the point of contact) is towards left.

*Multiple options can be correct
QUESTION: 16

The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed ω and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed ω/2. The ring and disc are separated by frictionless ball bearings. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of 30° with the horizontal. Then with respect to the horizontal surface,

Solution:

For rolling motion, the velocity of the point of contact with respect to the surface should be zero. For this

A shown in the figure, the point P will have two velocities

 

making an angle of 30º with the vertical due to rotation

*Multiple options can be correct
QUESTION: 17

Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

Solution:

The acceleration of the center of mass of cylinder rolling down an inclined plane is

Here IP > IQ because in case of P the mass is concentrated away from the axis.

*Multiple options can be correct
QUESTION: 18

In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angleθ with the horizontal floor. The coefficient of friction between the wall and the ladder is μ1 and that between the floor and the ladder is μ2. The normal reaction of the wall on the ladder is N1 and that of the floor is N2. If the ladder is about to slip, then

 

Solution:

When

Solving the above equation we get

∴ (c) is the correct option.
When μ1 = 0

Taking torque about P we get

∴ (d) is correct

*Multiple options can be correct
QUESTION: 19

A ring of mass M and radius R is rotating with angular speed ω about a fixed vertical axis passing through its centre O with two point masses each of mass M/8 at rest at O. Thesemasses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is 8/9 ω and one of the masses is
at a distance of 3/5 R from O. At this instant the distance of the other mass from O is

Solution:

Applying conservation of angular mumentum about the axis

D is the correct option

*Multiple options can be correct
QUESTION: 20

Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, rigid rod of length through their centres. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is ω. The angular momentum of the entire assembly about the point   (see the figure)Which of the following statement(s) is (are) true? 

Solution:

The circumference of a circle of radius OM will be 2π(5a) = 10πa.

For completing this circle once, the smaller disc will have to take

Therefore the C.M. of the assembly rotates about zaxis with an angular speed of w/5.
The angular momentum about the C.M. of the system

 

*Multiple options can be correct
QUESTION: 21

The position  of a particle of mass m is given by the following equation

where  which of the following statement(s) is(are) true about the particle?

 

Solution:

At t = 1s

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