Example 1:
In physics, the dynamics of rotation involve the study of the motion of objects that rotate about an axis. A common example is a spinning wheel or a rotating disk. Let’s consider the dynamics of a rotating disk to calculate its angular velocity and acceleration.
Problem:
Suppose we have a disk with a radius of 0.5 meters and a mass of 2 kg. The disk is initially at rest and experiences a constant torque of 10 Nm applied to its edge. Calculate the angular velocity and angular acceleration of the disk after 4 seconds.
Solution:
- Moment of Inertia (I): To calculate the angular acceleration, we first need to find the moment of inertia of the disk. For a solid disk, the moment of inertia is given by the formula:
I = (1/2) * M * R^2
where M is the mass of the disk (2 kg) and R is its radius (0.5 m).
I = (1/2) * 2 kg * (0.5 m)^2 = 0.25 kg * m^2
- Angular Acceleration (α): To find the angular acceleration, we can use the formula:
α = τ / I
where τ is the torque applied (10 Nm).
α = 10 Nm / 0.25 kg * m^2 = 40 rad/s^2
- Angular Velocity (ω): Since the disk is initially at rest, its initial angular velocity (ω₀) is 0 rad/s. We can use the formula for angular velocity under constant angular acceleration to find the final angular velocity after 4 seconds (t):
ω = ω₀ + α * t
ω = 0 rad/s + 40 rad/s^2 * 4 s = 160 rad/s
So, after 4 seconds, the angular acceleration of the disk is 40 rad/s^2, and its angular velocity is 160 rad/s.
Example 2:
- Angular Acceleration (α): To find the angular acceleration at any given time, we can use the formula:
α(t) = τ(t) / I
where I is the moment of inertia of the wheel (2 kg m²).
α(t) = (3t) / 2 kg m² = (3/2) * t rad/s²
- Angular Velocity (ω): Since the wheel is initially at rest, its initial angular velocity (ω₀) is 0 rad/s. To find the final angular velocity after 4 seconds (t), we need to integrate the angular acceleration function:
ω(t) = ω₀ + ∫α(t) dt
ω(t) = ∫(3/2) * t dt
We’ll integrate the function over the interval [0, 4]:
ω(4) = ∫(3/2) * t dt from 0 to 4
ω(4) = (3/2) * ∫t dt from 0 to 4
ω(4) = (3/2) * [ (1/2) * t² ] from 0 to 4
ω(4) = (3/2) * [(1/2) * (4)² – (1/2) * (0)²]
ω(4) = (3/2) * (8)
ω(4) = 12 rad/s
After 4 seconds, the angular velocity of the wheel is 12 rad/s.
Practice Problems
1. A spinning disk has a moment of inertia of 3 kg m². A motor applies a torque to the disk according to the function τ(t) = 4t² Nm, where t is the time in seconds. The disk is initially at rest. Calculate the angular velocity of the disk after 5 seconds.
2. A cylinder with a moment of inertia of 6 kg m² is initially rotating with an angular velocity of 10 rad/s. A torque is applied to the cylinder as a function of time, given by τ(t) = 8 – 2t Nm, where t is the time in seconds. Calculate the angular velocity of the cylinder after 3 seconds.
