Angular Acceleration
What is Angular Acceleration?
When you push a car, it accelerates in a straight line. This is "Linear Acceleration." But what happens when you spin a wheel, start a drill, or watch a figure skater spin?
In these cases, the object isn't going somewhere; it is spinning faster or slower. This change in rotational speed is called Angular Acceleration.
It is represented by the Greek letter Alpha (α). Just as linear acceleration measures how quickly your speed changes (meters per second squared), angular acceleration measures how quickly your spin rate changes (radians per second squared).
Method 1: Change in Velocity (Kinematics)
The most direct way to calculate angular acceleration is if you know how fast it was spinning at the start, how fast it is spinning at the end, and how long it took to change.
ω1 = Initial Angular Velocity (rad/s)
ω2 = Final Angular Velocity (rad/s)
t = Time taken (s)
Example:
A fan is spinning at 10 rad/s. You turn it up to high speed, and 5 seconds later, it is spinning at 30 rad/s.
Calculation: (30 - 10) / 5 = 4 rad/s².
Method 2: Newton's Second Law for Rotation
We all know Newton's famous law for straight lines: Force = Mass × Acceleration. But did you know there is an exact twin for rotation?
In the rotational world:
- Force becomes Torque (τ).
- Mass becomes Moment of Inertia (I).
- Acceleration becomes Angular Acceleration (α).
I = Moment of Inertia (kg·m²)
α (Alpha) = Angular Acceleration (rad/s²)
What is "Moment of Inertia"?
This is often the most confusing part for students. In linear motion, "Mass" determines how hard it is to push something. In rotation, "Inertia" determines how hard it is to spin something.
Crucially, inertia depends not just on how heavy the object is, but where the weight is located.
If the weight is on the edges (like a bicycle wheel), it is hard to start spinning (High Inertia).
If the weight is in the center (like a solid marble), it spins easily (Low Inertia).
Linear vs. Angular: The Connection
Physics is beautiful because these two worlds are connected. If you know how fast a wheel is accelerating angularly ($\alpha$), and you know the radius of the wheel ($r$), you can find out the linear acceleration ($a$) of a point on the edge of the tire.
Formula: a = r × α
This is how electric cars with high-torque motors (creating high angular acceleration at the wheels) achieve massive 0-60 mph times.
Comparison Table: Linear vs. Rotational
Use this cheat sheet to translate your linear physics problems into rotational ones.
| Property | Linear Variable | Rotational Variable |
|---|---|---|
| Position | Distance (d) | Angle (θ) |
| Speed | Velocity (v) | Angular Velocity (ω) |
| Speed Change | Acceleration (a) | Angular Acceleration (α) |
| Resistance | Mass (m) | Moment of Inertia (I) |
| Cause | Force (F) | Torque (τ) |
Real World Example: The Figure Skater
A classic example of these physics concepts is a figure skater spinning.
- The skater starts spinning with arms out. They have a High Moment of Inertia (mass is far from center). Their velocity ($\omega$) is slow.
- They pull their arms in tight. They have changed their shape, lowering their Moment of Inertia.
- Because Angular Momentum is conserved, when Inertia ($I$) drops, Velocity ($\omega$) MUST skyrocket. The skater suddenly spins incredibly fast without pushing off the ice again.
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