Rotational dynamic

Most Important Previous Year Questions
Rotational Dynamics

2024, 2022, 2019, 2016

Frequency: Asked 4 times in last 8 years

Question 1: Angular Momentum Conservation (5 marks)

A platform of radius 2m and mass 100kg is rotating with angular velocity of 10 rad/s. A person of mass 60kg standing at the center walks to the edge of the platform. Calculate:

1. Initial angular momentum of the system
2. Final angular velocity of the platform-person system
3. Change in rotational kinetic energy

Solution:

Let's solve this step by step:

1. Initial angular momentum (Li):

Li = I₁ω₁ = (½MR²)ω₁

Li = ½(100)(2²)(10) = 2000 kg⋅m²/s

2. Final moment of inertia:

If = ½(100)(2²) + (60)(2²) = 440 kg⋅m²

3. Final angular velocity (by conservation of angular momentum):

2000 = 440ωf

ωf = 4.55 rad/s

4. Change in rotational KE = ½If(ωf²) - ½Ii(ωi²)

2023, 2020, 2017, 2014

Frequency: Asked 4 times in last 10 years

Question 2: Moment of Inertia (4 marks)

Derive the moment of inertia of a solid cylinder about its axis using integration. How does the moment of inertia change when the cylinder is hollow?

Solution:

For solid cylinder:

1. Consider a cylindrical shell of radius r and thickness dr

2. dI = r²dm where dm = ρ(2πr)(h)dr

3. I = ∫dI = ∫r²ρ(2πr)(h)dr from 0 to R

4. I = ½MR² where M is total mass

For hollow cylinder:

I = ½M(R₁² + R₂²) where R₁ and R₂ are outer and inner radii

2023, 2021, 2018, 2015, 2012

Frequency: Asked 5 times in last 12 years

Question 3: Torque and Angular Acceleration (5 marks)

A wheel of moment of inertia 0.5 kg⋅m² is acted upon by a torque that varies with time according to the relation τ = 3t² - 4t + 2 (where τ is in N⋅m and t in seconds). Find the angular acceleration at t = 2s and the angular velocity at t = 2s if the wheel starts from rest.

Solution:

1. Torque equation: τ = 3t² - 4t + 2

2. Angular acceleration: α = τ/I

3. At t = 2s:

τ = 3(2)² - 4(2) + 2 = 12 - 8 + 2 = 6 N⋅m

α = 6/0.5 = 12 rad/s²

4. For angular velocity:

ω = ∫α dt = (1/I)∫(3t² - 4t + 2)dt

ω = (1/0.5)(t³ - 2t² + 2t) from 0 to 2

Final ω = 8 rad/s