Rotational Motion MCQ Quiz - Objective Question with Answer for Rotational Motion - Download Free PDF

Calculation:
W_F = 20 × 1 = 20 J

Therefore, ΔKE = 20 J = (1/2) I ω²

I = M R² = 10 × 0.1² = 0.1 kg m²

Therefore, 20 = (1/2) × 0.1 × ω²

Implying, ω = 20 rad/sec

Solution:

Consider the disc and the rings together as a system. Since external torque is zero, angular momentum of the system about the vertical axis through point O is conserved.

Ii × ωi = If × ωf      (1)

Initial moment of inertia (Ii):

For a disc, Ii = (1/2) × M × R² = (1/2) × 50 × (0.4)² = 4 kg·m²

Final moment of inertia (If):

The moment of inertia of each ring about the rotation axis using parallel axis theorem:

Iring = m × r² + m × d² = 2 × m × r² (since d = r)

So for two rings: Irings = 2 × (6.25) × (0.2)² = 0.5 kg·m²

Total If = Idisc + Irings = 4 + 1 = 5 kg·m²

Substitute in equation (1):

4 × 10 = 5 × ωf ⇒ ωf = 8 rad/s

Answer: 8 rad/s

Correct option is: (3) 7 / 57

For a larger solid sphere about diameter Y-axis:

I_whole = (2 / 5) × M × (2R)² = (8 / 5) × M × R²

Density of sphere is uniform:

M / V_whole = M_smaller / V_smaller

M / ((4/3)π(2R)³) = M′ / ((4/3)πR³)

⇒ M′ = M / 8

Using parallel axis theorem for smaller sphere:

I′ = I_cm + M′ × R² = (2 / 5) × (M / 8) × R² + (M / 8) × R² = (7 / 40) × M × R²

Ratio:

Ratio = I_smaller / I_remaining = I′ / (I_whole − I′)

= ((7 / 40) × M × R²) / (((8 / 5) − (7 / 40)) × M × R²)

= 7 / 57

Calculation:

For translational equilibrium

N₁ = Mg

N₂ = f

For rotational equilibrium

Torque about A, MgL/2 cosθ = N₂L sinθ

(Mg/2) cotθ = N₂ = f

(Mg/2) cot 30° = f

(Mg/2) √3 = N₂

100√3 = f

Calculation:
Case I: mR²/2 ω₀ = (mR²/2 + mR²/4 + mR²/4) ω ω = ω₀/2 = 3 rad/s

Case II: ω = ω₀ = 6 rad/s

Case III: mR²/2 ω₀ + mvR²/2 = (mR²)ω So, 3 + 8/2 = 7 rad/s

Case IV: mR²/2 ω₀ + mvR²(1/2) = mR²ω 3 + 8/4 = 5 rad/s

The correct option is: 2

Concept Used:

• Work-Energy Theorem: This theorem states that the net work done by the force on a body is equal to the change in its kinetic energy.
• Mathematically, Work Done (W) = Final Kinetic Energy - Initial Kinetic Energy
• Kinetic Energy (K.E) is given by the formula: K.E = (1/2) × m × v²
• So, W = (1/2) × m × (v² - u²), where:
  • m = mass of the object
  • v = final velocity
  • u = initial velocity

Calculation:

• Given:
  • Mass of the car (m) = 1500 kg
  • Initial velocity (u) = 36 km/h = 36 × (1000 ÷ 3600) = 10 m/s
  • Final velocity (v) = 72 km/h = 72 × (1000 ÷ 3600) = 20 m/s
• Using the work-energy theorem:
  W = (1/2) × m × (v² - u²)
  ⇒ W = (1/2) × 1500 × (20² - 10²)
  ⇒ W = (1/2) × 1500 × (400 - 100)
  ⇒ W = (1/2) × 1500 × 300
  ⇒ W = 750 × 300
  ⇒ W = 225000 J
  ⇒ W = 2.25 × 10⁵ J

Additional Information:

• Work done is a scalar quantity and is expressed in joules (J) in the SI system.
• This type of question is commonly asked in mechanics under the topic of energy and work.
• Always convert velocities from km/h to m/s before substituting in the kinetic energy formula: multiply by (1000 ÷ 3600) or simplify as 5/18.

Diagram/Visual Aid Suggestion:

• A diagram showing a car moving initially at a lower speed and then at a higher speed with vectors labeled u and v.
• Include a bar graph representing the initial and final kinetic energy for visual comparison.

CONCEPT:

• Rotational motion: When a block is moving about a fixed axis on a circular path then this type of motion is called rotational motion.

Torque (τ): 

• It is the twisting force that tends to cause rotation.
• The point where the object rotates is known as the axis of rotation. 
• Mathematically it is written as,

τ = rFsin θ 

EXPLANATION:

• The power associated with torque is given by the product of torque and angular velocity of the body about an axis of rotation i.e.,

⇒ P = τω 

Where τ = torque and ω = angular velocity 

EXTRA POINTS:

In rotational dynamics

• Moment of inertia is the analogue of mass
• Angular velocity is analogue of linear velocity
• Angular acceleration is analogue of linear acceleration

Thus, in linear motion mass x velocity = momentum

The analogues of the moment of inertia x angular velocity = angular momentum

The correct answer is option 4) i.e. L = √(2EI)

CONCEPT:

• Angular momentum: The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity.
  • Angular momentum also obeys the law of conservation of momentum i.e. angular momentum before and after is conserved.

Angular momentum, L = I × ω 

Rotational kinetic energy: For a given fixed axis of rotation, the rotational kinetic energy is given by:

 \(KE = \frac{1}{2} Iω^2\)

Where I is the moment of inertia, ω is the angular velocity.

CALCULATION:

Angular momentum, L = I × ω      ----(1) 

Roational kinetic energy = \(E = \frac{1}{2} Iω^2\) ⇒ ω = \(\sqrt{\frac{2E}{I}}\)      ----(2)

Substituting (2) in (1) we get

L = I × \(\sqrt{\frac{2E}{I}}\)= √(2EI)

CONCEPT:

• The angular momentum of a particle rotating about an axis is defined as the moment of the linear momentum of the particle about that axis.
• It is measured as the product of linear momentum and the perpendicular distance of its line of action from the axis of rotation.
• The relation between the angular momentum and moment of inertia is given by

L = Iω

Where I = moment of inertia, L = angular momentum, and ω = angular velocity.

EXPLANATION:

• If there is no external torque acting on system then initial angular momentum (L_initial) of system is equal to final momentum (L_final).
• Hence, the angular momentum in a closed system is a conserved.

∴ Iω = constant

⇒ I ∝ 1/ω

i.e. Moment of inertia is inversely proportional to the angular velocity.

• Hence if the moment of inertia of a rotating body is increased then the angular velocity decreases.

CONCEPT:

• Angular acceleration (α): It is defined as the time rate of change of angular velocity of a particle is called its angular acceleration.
• If Δω is the change in angular velocity time Δt, then average acceleration is

\(\vec α = \frac{{{\rm{\Delta }}\omega }}{{{\rm{\Delta }}t}}\)

Moment of Inertia (I):

• Moment of inertia plays the same role in rotational motion as mass plays in linear motion. It is the property of a body due to which it opposes any change in its state of rest or of uniform rotation.
• Moment of inertia of a particle is

I = mr2

where r = perpendicular distance of the particle from the rotational axis.

Torque (τ): 

• It is the twisting force that tends to cause rotation.
• The point where the object rotates is known as the axis of rotation. 
• Mathematically it is written as,

τ = rFsin θ 

EXPLANATION:

The relationship between the angular acceleration (α), torque (τ) and moment of inertia (I) is given by

⇒ τ = α × I

\( \Rightarrow \alpha = \frac{\tau }{I}\)

• Therefore, angular acceleration = Torque / Moment of inertia. Hence option 2 is correct.

CONCEPT:

• Rotational kinetic energy: The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
• A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
• Mathematically rotational kinetic energy can be written as -

\(KE = \frac{1}{2}I{ω ^2}\)

Where I = moment of inertia and ω = angular velocity.

EXPLANATION:

• From above it is clear that, \(\frac{1}{2}I{ω ^2}\) represents rotational kinetic energy.

CONCEPT:

Angular momentum (L):

• The angular momentum of a rigid body is defined as the product of the moment of inertia and the angular velocity i.e.,

\(⇒ L = Iω \)

Where I = moment of inertia and ω = angular velocity

Law of conservation of angular momentum:

• When the net external torque acting on a body about a given axis is zero, the total angular momentum of the body about that axis remains constant i.e.,

⇒ I1ω1 = I2ω2

EXPLANATION:

• When a man standing on a revolving platform and spreading his hands suddenly as he stretches his hand that he is increasing his moment of inertia thereby decreasing his angular velocity using the principle of conservation of angular momentum. Therefore option 2 is correct.

CONCEPT:

Torque (τ): 

• It is the twisting force that tends to cause rotation.
• The point where the object rotates is known as the axis of rotation. 
• Mathematically it is written as,

τ = rFsin θ 

• Angular acceleration (α): It is defined as the time rate of change of angular velocity of a particle is called its angular acceleration.
• If Δω is the change in angular velocity time Δt, then average acceleration is

\(\vec α = \frac{{{\rm{\Delta }}\omega }}{{{\rm{\Delta }}t}}\)

EXPLANATION:

• Torque is the measure of the amount of force acting on an object that can cause it to rotate.
• The torque that is needed to produce angular acceleration depends on the mass distribution of the object which is described by the moment of inertia.
• Therefore, torque (\(\tau \)) is the product of the angular acceleration (a) and the moment of inertia (I) of an object. Therefore option 2 is correct.

           \(\tau = I \times a\)

CONCEPT:

Moment of inertia:

• The moment of inertia of a rigid body about a fixed axis is defined as the sum of the product of the masses of the particles constituting the body and the square of their respective distances from the axis of the rotation.
• The moment of inertia of a particle  is

⇒ I = mr2

Where r = the perpendicular distance of the particle from the rotational axis.

• Moment of inertia of a body made up of a number of particles (discrete distribution)

⇒ I = m1r12 + m2r22 + m3r32 + m4r42 + -------

Rotational kinetic energy: 

• The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
• A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
• Mathematically rotational kinetic energy can be written as -

⇒ KE \( = \frac{1}{2}I{\omega ^2}\)

Where I = moment of inertia and ω = angular velocity.

EXPLANATION:

• The moment of inertia of the ring about an axis passing through the center and perpendicular to its plane is given by

⇒ I_ring = MR²

• Moment of inertia of the disc about an axis passing through center and perpendicular to its plane is given by -

\(⇒ {I_{disc}} = \frac{1}{2}M{R^2}\)

• As we know that mathematically rotational kinetic energy can be written as

\(⇒ KE = \frac{1}{2}I{\omega ^2}\)

• According to the question, the angular velocity of a thin disc and a thin ring is the same. Therefore, the kinetic energy depends on the moment of inertia.
• Therefore, a body having more moments of inertia will have more kinetic energy and vice - versa.
• So, from the equation, it is clear that,

⇒ Iring > Idisc

∴ Kring > Kdisc

• The ring has higher kinetic energy.

Concept:

The equations which are applied in linear motion can also be applied in angular motion.

Calculation:

Given:

α = 3 rad/s², ω_o = 2 rad/s, t = 2 s

We know that,

\(θ = {ω _0}t + \frac{1}{2}α {t^2}\)

\(θ = ({2}\times 2) + \frac{1}{2}\times3\times2^2\)

θ = 4 + 6 = 10 radians