Test: Angular Acceleration - JEE MCQ

The period of a particle in circular motion is the time taken to complete one full revolution. The period is given by T = 2π/ω, where ω is the angular velocity. Angular acceleration, α, does not affect the period directly. The angular acceleration determines how quickly the angular velocity changes, but it does not affect the time taken for one complete revolution. Therefore, if the angular acceleration is doubled, the period of the particle's motion will remain the same.

CONCEPT:

• Angular velocity (ω): How fast the angular position or orientation of an object changes with time or How fast an object rotates or revolves relative to another point is known as angular velocity.
  • Angular velocity in terms of RPM (revolutions per minute) 

ω = 2 π f / 60

where ω is in terms of radian per second and f is in revolution per minute.

• Angular acceleration (α): The rate of change of angular velocity is known as angular acceleration.

The relation between α and changing ω is given by: 

ω₁ = ω₀ + α t

where ω₁ is the final angular velocity, ω₀ is the initial angular velocity α is the angular acceleration.

CALCULATION:

Given that f₀ = 2100 rpm; f₁ = 0 rpm; t = 2 sec.

ω₀ = 2 π f₀ / 60 = 2 π 2100 / 60 = 70π 

ω₁ = 0

ω₁ = ω₀ + α t

0 = 70π + α 2

α = -35π rad/sec²

So the correct answer is option 1.

CONCEPT:

• Angular displacement (θ): The angular displacement of a body is the angle in degrees or radians or revolutions through which a point revolves around a point or axis.
• Angular velocity (ω): How fast the angular position or orientation of an object changes with time or How fast an object rotates or revolves relative to another point is known as angular velocity.
  • It is the rate of change of angular displacement.​
• Angular acceleration (α): The rate of change of angular velocity is known as angular acceleration.
• Angular momentum (L): The moment of momentum is known as angular momentum.

CALCULATION:

• The rate of change of angular velocity is known as angular acceleration. 

Mathematically,

α = dω/dt

• So the correct answer is option 3.

Kinematic equations of rotational motion about a fixed axis,

where ω is the instantaneous angular velocity, α is the instantaneous angular acceleration, θ is the angular displacement, and t is time.

EXPLANATION:

Given: Length of the rod = L, and angular displacement θ = 2t²

• Using the first equation,

• Using the second equation, 

Therefore option 4 is correct.

Moment of inertia:

• 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 + -------

EXPLANATION:

Given: mass of the particle = m, distance from x-axis = b.

The moment of inertia for the particle of mass m from an axis at a distance b is,

⇒ I = mb²

Therefore option 3 is correct.

Angular acceleration is defined as the rate at which an object's angular velocity changes over time. It is given by the formula:
Angular acceleration (α) = Change in angular velocity (Δω) / Change in time (Δt)

To find the time it takes for the disc to come to a complete stop, we can use the formula:
ω = ω_₀ + αt

where:
ω = final angular velocity (0 rad/s, as the disc comes to a complete stop),
ω_₀ = initial angular velocity (200 rad/s),
α = angular acceleration, and
t = time.

Rearranging the formula to solve for time (t):
t = (ω - ω_₀) / α

Substituting the given values:
t = (0 - 200 rad/s) / (-10 rad/s²)
t = 20 seconds

Therefore, it will take 20 seconds for the disc to come to a complete stop.

The number of revolutions (N) completed by a wheel can be calculated using the formula:
N = (ω_₀t + (1/2)αt²) 

where:
ω₀ = initial angular velocity (0 rad/s, as the wheel starts from rest),
t = time, and
α = angular acceleration.

Simplifying the formula for the given scenario:
N = (1/2)αt² 

Substituting the given values:
N = (1/2) * 4 rad/s² * (10 s)² 
N = 200 revolutions

Therefore, the wheel will complete 200 revolutions in 10 seconds.