CBSE Class 12 Physics Rotational Motion Exam Notes

ROTATIONAL KINEMATICS

A rigid body will be having rotational motion about an axis of rotation when each particle of the body moves in a circular path around their centres which lie along a straight line on the axis of rotation. The angular velocity, ω, is given by

The relations between linear displacement, s, velocity, v, and acceleration, a, to the quantities describing circular motion in rotational kinematics i.e., angular displacement θ, angular velocity, ω, and angular acceleration, α, respectively are given for distance r from axis of rotation as:

MOMENT OF INERTIA

There are two types of systems discrete and continuous.
Discrete System: The moment of inertia of a body is a measure of its rotational inertia. The moment of inertia of a body about an axis is defined as the sum of the products of the masses of the particles constituting a body and the squares of the respective distances from the axis of rotation. If a body consists of masses m1, m2, m3......... at perpendicular distances r1, r2, r3....... respectively from axis of rotation, the moment of inertia, I, is given by

THEOREMS OF MOMENT OF INERTIA

(a) Theorem of Parallel Axes : It states that the moment of inertia of a body about an axis is equal to the sum of moment of inertia about a parallel axis through its centre of mass and product of mass of the body and square of perpendicular distance between two axes. If the moment of inertia of a body about its centre of mass and through axis AB perpendicular to its plane is ICM, then the moment of inertia of this body through any axis CD parallel to this axis, I_parallel, is given by   I_parallel = I_CM + Md² D 

 where d is the perpendicular distance between the two axes

(b) Theorem of Perpendicular Axes : The sum of moments of inertia of a body about two axes Ix and Iy at right angles to each other in the plane of the body passing through its centre of masses is equal to the moment of inertia of the body Iz about an axis perpendicular to these axes but passing through their point of interaction at the centre of mass.  Iz = Ix + Iy

MOMENT OF INERTIA OF SOME CONTINUOUS SYSTEM

(i) M.I. OF A ROD
Case-I: About an axis which is passing through the C.M. and ⊥r to its length:
Let us take a small element of thickness dx, whose mass is dm.

Question. A grind stone starts from rest and has a constant angular acceleration of 4.0 rad/sec². The angular displacement and angular velocity, after 4 sec. will respectively be:
a. 32 rad, 16 rad / sec
b. 16 rad, 32 rad / sec
c. 64 rad, 32 rad / sec
d. 32 rad, 64 rad / sec
Answer : A

Question. A stone tied to string is rotated in a vertical circle. The minimum speed with which the string has to be rotated:
a. Decreases with increasing mass of the stone
b. Is independent of the mass of the stone
c. Decreases with increasing in length of the string
d. Is independent of the length of the string Conical Pendulum
Answer : B

Question. A string of length L is fixed at one end and carries a mass M at the other end. The string makes 2/π revolutions per second around the vertical axis through the fixed end as shown in the figure, then tension in the string is:

a. ML
b. 2 ML
c. 4 ML
d. 16 ML
Answer : D

Question. Five particles of mass = 2 kg are attached to the rim of a circular disc of radius 0.1 m and negligible mass. Moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is:
a. 1 kg m²
b. 0.1 kg m²
c. 2 kg m²
d. 0.2 kg m²
Answer : B

Question. If the equation for the displacement of a particle moving on a circular path is given by (θ ) = 2t³ + 0.5, where θ is in radians and t in seconds, then the angular velocity of the particle after 2 sec from its start:
a. 8 rad / sec
b. 12 rad / sec
c. 24 rad / sec
d. 36 rad / sec
Answer : C

Question. The minimum speed for a particle at the lowest point of a vertical circle of radius R, to describe the circle is ‘v’. If the radius of the circle is reduced to one-fourth its value, the corresponding minimum speed will be:
a. v/4
b. v/2
c. 2v
d. 4v
Answer : B

Question. A body slides down a frictionless track which ends in a circular loop of diameter D, then the minimum height h of the body in term of D so that it may just complete the loop, is:
a. h = 5D/2
b. h = 5D/4
c. h = 3D/4
d. h = D/4
Answer : B

Question. Two circular discs A and B are of equal masses and thickness but made of metals with densities d_A and dB (d_A > d_B). If their moments of inertia about an axis passingthrough centres and normal to the circular faces be I_A and I_B , then:
a. I_A = I_B
b. I_A > I_B
c. I_A < I_B
d. I_A > = < I_B
Answer : C

Question. A horizontal heavy uniform bar of weight W is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to:

a. W
b. W./2
c 3/W
d. W/4
Answer : D

Question. The angular velocity of seconds hand of a watch will be:
a. (π/60) rad / sec
b. (π/30) rad / sec
c. 60π rad / sec
d. 30π rad / sec
Answer : B

Question. If the position vector of a particle is r = (3iˆ + 4 ˆj) meter and its angular velocity isω = ( ˆj + 2kˆ) rad/sec then its linear velocity is: (in m/s)
a. (8iˆ −6 ˆj + 3kˆ)
b. (3iˆ + 6 ˆj +8kˆ)
c. −(3iˆ + 6 ˆj + 6kˆ)
d. (6iˆ +8 ˆj + 3kˆ)
Answer : A

Question. A wheel completes 2000 rotations in covering a distance of 9.5 km . The diameter of the wheel is:
a. 1.5m
b. 1.5cm
c. 7.5m
d. 7.5cm
Answer : A

Question. A wheel is at rest. Its angular velocity increases uniformly and becomes 60 rad/sec after 5 sec. The total angular displacement is:
a. 600rad
b. 75rad
c. 300rad
d.150 rad
Answer : D

Question. A force of (2iˆ − 4 ˆj + 2kˆ)N acts at a point (3iˆ + 2 ˆj − 4kˆ) metre from the origin. The magnitude of torque is:
a. Zero
b. 24.4 N-m
c. 0.244 N-m
d. 2.444 N-m
Answer : B

Question. A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity ω₀. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform ω (t) will vary with time t as:

Answer : B

Question. Moment of inertia of a uniform circular disc about a diameter is I. Its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim will be:
a. 5 I
b. 6 I
c. 3 I
d. 4 I
Answer : B

Question. The moment of inertia of a solid sphere of density ρ and radius R about its diameter is:
a. (105/176) R⁵p
b. (105/176) R²p
c. (176/105) R⁵p
d. (176/105) R²p
Answer : C

Question. A particle is moving along a circular path of radius 3 meter in such a way that the distance travelled measured along the circumference is given by S = t²/2 + t³/3  The acceleration of particle when t = 2 sec is:
a. 1.3 m/s²
b. 13 m/s²
c. 3 m/s²
d. 10 m/s²
Answer : B

Question. An electric fan is rotating at a speed of 600 rev/minute. When the power supply is stopped, it stops after 60 revolutions. The time taken to stop is:
a. 12 s
b. 30 s
c. 45 s
d. 60 s
Answer : A

Question. A grinding wheel attained a velocity of 20 rad/sec in 5 sec starting from rest. Find the number of revolutions made by the wheel:
a. (π/25) rev / sec
b. (1/π) rev/sec
c. (25/π) rev/sec
d. None of these
Answer : C

Question. The angular velocity of a particle is given by ω = 1.5 t − 3t² + 2, the time when its angular acceleration ceases to be zero will be:
a. 25 sec
b. 0.25 sec
c. 12 sec
d. 1.2 sec
Answer : B

Question. The resultant of the system in the figure is a force of 8N parallel to the given force through R. The value of PR equals to:

a. (1/4) RQ
b. (3/8) RQ
c. (3/5) RQ
d. (2/5) RQ
Answer : C

Question. The position of a particle is given by: r = (iˆ + 2 ˆj − kˆ) and momentum P = (3iˆ + 4 ˆj − 2kˆ).
The angular momentum is perpendicular to:
a. X-axis
b. Y-axis
c. Z-axis
d. Line at equal angles to all the three axes
Answer : A

Question. A body of moment of inertia of 3 kg-m² rotating with an angular velocity of 2 rad/sec has the same kinetic energy as a mass of 12 kg moving with a velocity of:
a. 8 m/s
b. 0.5 m/s
c. 2 m/s
d. 1 m/s
Answer : D

Question. A disc and a ring of same mass are rolling and if their kinetic energies are equal, then the ratio of their velocities will be:
a. √4 : √3
b. √3 : √4
c. √3 : √2
d. √2 : √3
Answer : A

Question. A solid cylinder of mass M and radius R rolls without slipping down an inclined plane of length L and height h. What is the speed of its centre of mass when the cylinder reaches its bottom?

Answer : B

Question. Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a light rigid rod. If an impulse J = Mv is imparted to the body at one of its ends, what would be its angular velocity:

a. v/L
b. 2v/L
c. v/3L
d. v/4L
Answer : A

Question. A ring of radius 0.5 m and mass 10 kg is rotating about its diameter with an angular velocity of 20 rad/s. Its kinetic energy is:
a. 10 J
b. 100 J
c. 500 J
d. 250 J
Answer : D

Question. An automobile engine develops 100 kW when rotating at a speed of 1800 rev/min. What torque does it deliver?
a. 350 N-m
b. 440 N-m
c. 531 N-m
d. 628 N-m
Answer : C

Question. A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and moment of inertia about it is I. A weight mg is attached to the end of the cord and falls from rest. After falling through a distance h, the angular velocity of the wheel will be:

Answer : B

Question. In the following figure, a body of mass m is tied at one end of a light string and this string is wrapped around the solid cylinder of mass M and radius R. At the moment t = 0 the system starts moving. If the friction is negligible, angular velocity at time t would be:

a. mgRt / (M + m)
b. 2Mgt / (M + 2m)
c. 2mgt / R(M − 2m)
d. 2mgt / R(M + 2m)
Answer : D

Question. A solid sphere and a disc of same mass and radius starts rolling down a rough inclined plane, from the same height the ratio of the time taken in the two cases is:
a. 15 : 14
b. √15 : √14
c. 14 : 15
d. √14 : √15
Answer : D

Question. The angular velocity of a body is ω = 2iˆ +3ˆj + 4kˆ and a torque τ = iˆ + 2 ˆj +3kˆ  acts on it. The rotational power will be:
a. 20 W
b. 15 W
c. √17 W
d. v14 W
Answer : A

Question. A flywheel of moment of inertia 0.32 kg-m² is rotated steadily at 120 rad / sec by a 50W electric motor. The kinetic energy of the flywheel is:
a. 4608 J
b. 1152 J
c. 2304 J
d. 6912 J
Answer : C

Question. A sphere rolls down on an inclined plane of inclination θ.
What is the acceleration as the sphere reaches bottom:
a. (5/7)g sin θ
b.(3/5)g sin θ
c. (2/7)g sin θ
d. (2/5)g sin θ
Answer : A

Question. A block with a square base measuring a×a and height h, is placed on an inclined plane. The coefficient of friction is μ.The angle of inclination (θ ) of the plane is gradually increased. The block will:

a. topple before sliding if μ > a/h
b. topple before sliding if μ < a/h
c. slide before toppling if μ > a/h
d. slide before toppling if μ < a/h
Answer : A, D

Question. A ring rolls without slipping on the ground. Its center C moves with a constant speed u. P is any point on the ring. The speed of P with respect to the ground is υ , then:
a. 0 ≤ υ ≤ 2u
b. υ = u, if CP is horizontal
c. υ = u, if CP makes an angle of 60º with the horizontal and P is below the horizontal of C
d. υ = √2u, if CP is horizontal
Answer : A, C, D

Question. The moments of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the center O and perpendicular to the plate are: (where I₁, I₂, I₃ and I₄ are respectively the moments of inertia about axes 1, 2, 3 and 4, where axes are in the plane of the plate)
a. I₁ + I₂
b. I₃ + I₄
c. I₁ + I₄
d. I₁ + I₂ + I₃ + I₄
Answer : A, B, C

Question. 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 speeds 2υ and υ respectively strike the bar (as shown in figure) and stick to the bar after collision. Denoting angular velocity (about center of mass) total energy and velocity of center of mass by ω , E and υ₀ respectively, we have after collision:

a. υ₀ = 0
b. ω = 3υ / 5a
c. ω = υ / 5a
d. E = 3/5 mυ²
Answer : A, C, D

Question. In a free space, a rifle of mass M shoots a bullet of mass m at a stationary block of mass M distance D away from it. When the bullet has moved through a distance d towards the block, the center of mass of the bullet block system is at a distance of:
a. m(D - d) / M + m from the block
b.  (m + M)d / M from the rifle
c. M(D + d) / M + m from the bullet d. none of the above
Answer : A, C

Question. The end B of the rod AB which makes angle θ with the floor is being pulled with a constant velocity υ₀ as shown.
The length of the rod is l. At the instant with θ = 37°;

a. velocity of end A is 4/3 υ₀ downward
b. angular velocity of the rod is 5υ₀/3 l
c. angular velocity of rod is constant
d. velocity of end A is constant
Answer : A, B

Question. A wheel of radius r rolls without slipping with a speed v on a horizontal road. When it is at a point A on the road, a small blob of mud separates form the wheel at its highest point and lands at a point B on the road. Which of the following are not correct?
a. AB = υ √r / g
b. AB = 2υ √r / g
c. AB = 4υ √r / g
d. If υ > √4rg, the blob of mud will land on the wheel and not on the road.
Answer : A, B, D

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