5.1 Rotational Kinematics
Just as objects can translate (move in straight lines), they can also rotate (spin around an axis). To analyze this, we translate our familiar linear kinematics terms into their rotational equivalents.
| Concept | Linear (Translational) | Rotational (Angular) |
|---|---|---|
| Position / Displacement | x or Δx (meters) | θ or Δθ (radians) |
| Velocity | v (m/s) | ω (rad/s) |
| Acceleration | a (m/s2) | α (rad/s2) |
| Time | t (seconds) - Time is the same in both! | |
The Rotational Kinematic Equations
If angular acceleration (α) is constant, we can use the rotational versions of the "Big Three" kinematic equations. They function identically to their linear counterparts.
5.2 Connecting Linear and Rotational Motion
How fast is a specific point on a spinning object moving through space? The linear (tangential) motion of a point depends on how fast the object is rotating and how far that point is from the axis of rotation.
The Bridge Equations: These equations connect the linear world to the rotational world via the radius (r).
Arc Length (Distance)
Tangential Velocity
Tangential Acceleration
5.3 Torque
Forces cause linear acceleration. Torque (τ) causes angular acceleration. Torque is the rotational equivalent of force; it is a measure of how effectively a force can cause an object to twist or spin.
Torque Formula
- τ = Torque (N·m)
- r = Lever arm (distance from axis to where force is applied)
- F = Magnitude of the force
- θ = Angle between the lever arm and force vector
⚠️ Direction Convention:
By standard physics convention, forces that cause Counter-Clockwise (CCW) rotation produce Positive (+) Torque. Forces that cause Clockwise (CW) rotation produce Negative (-) Torque.
5.4 Rotational Inertia
Mass is a measure of an object's resistance to linear acceleration. Rotational Inertia (I), also called the moment of inertia, is a measure of an object's resistance to angular acceleration.
Rotational inertia depends on two things: the total mass of the object, and how that mass is distributed relative to the axis of rotation.
The further the mass is from the axis of rotation, the harder it is to start or stop spinning (higher I).
Thought Experiment: The Figure Skater
A spinning ice skater pulls their arms in tightly. They haven't lost any mass, but they moved their mass closer to the axis of rotation (decreasing 'r'). This drastically reduces their rotational inertia (I), making them much easier to spin (which we'll see in Unit 6 causes their angular velocity to increase!).
5.5 Rotational Equilibrium and Newton's First Law in Rotational Form
Just as translational equilibrium means ΣF = 0, rotational equilibrium means that the torques are perfectly balanced.
Rotational Equilibrium: When the sum of all torques acting on an object is zero, the object will have no angular acceleration (α = 0). It will either remain at rest or continue to rotate at a constant angular velocity.
🎯 The Arbitrary Axis Trick:
If an object is in static equilibrium (not moving at all), it is in equilibrium relative to every possible axis. You can choose to place your "pivot point" exactly where an unknown force acts. Since r=0 at that point, that unknown force creates zero torque and drops out of your equation!
5.6 Newton's Second Law in Rotational Form
We arrive at the master equation for this unit. We swap out the translational variables in a = ΣF/m for their rotational counterparts to define how an object angularly accelerates.
The angular acceleration of an object is directly proportional to the net torque and inversely proportional to its rotational inertia.
Unit 5 Key Takeaways
Kinematics equations work exactly the same in a circle, just use θ, ω, and α.
Use v = rω and a = rα to bridge translational and rotational worlds.
Torque (τ = rF sinθ) requires a force applied at a distance from a pivot.
Rotational Inertia (I) increases drastically as mass is moved further from the axis.
In equilibrium, clockwise torques perfectly cancel out counter-clockwise torques.
Net Torque equals Rotational Inertia times Angular Acceleration (Στ = Iα).
End of Unit 5 Study Guide.