Rotational Motion – Angular Velocity, Acceleration & Energy Explained | Chapter 9 of University Physics

Chapter 9 of University Physics extends linear kinematics and dynamics into rotational systems for rigid bodies. You’ll learn how to describe angular position, velocity, and acceleration; link rotation to linear motion; compute moments of inertia; and apply energy methods to rotating systems.

Rotational Kinematics

Analogous to linear motion, rotational kinematics uses angular variables:

• Angular displacement: Δθ = θ₂ – θ₁
• Average angular velocity: ω_avg = Δθ / Δt
• Instantaneous angular velocity: ω = dθ/dt
• Angular acceleration: α = dω/dt = d²θ/dt²

Angular velocity is a vector aligned with the rotation axis (by the right-hand rule), while angular speed is its magnitude.

Constant Angular Acceleration

For constant α, the rotational analogs of linear equations are:

• ω = ω₀ + α t
• θ = θ₀ + ω₀ t + ½ α t²
• ω² = ω₀² + 2 α (θ – θ₀)

Linking Rotation to Linear Motion

A point at distance r from the axis moves with:

• Linear velocity: v = r ω
• Tangential acceleration: a_tan = r α
• Radial (centripetal) acceleration: a_rad = ω² r = v² / r

These relations connect rotational motion analyses to everyday linear measurements.

Moment of Inertia & Rotational Kinetic Energy

The moment of inertia (I) quantifies a body’s resistance to angular acceleration:

• Discrete masses: I = Σ m_i r_i²
• Continuous bodies: I = ∫ r² dm

Rotational kinetic energy parallels its linear form:

K_rot = ½ I ω²

Parallel-Axis Theorem

To find I about any axis a distance d from the center of mass:

I_p = I_cm + M d²

This theorem simplifies calculating inertia for off-center rotations.

Energy Conservation in Rotational Systems

Rotational energy contributes to total mechanical energy. When no nonconservative torques act:

½ I ω₁² + U = ½ I ω₂² + U

Here, U may include gravitational or elastic potential terms for rotating objects in a field or attached to springs.

Conclusion

Chapter 9 equips you with the tools to analyze any rigid body rotation—from spinning wheels to rolling objects—using angular analogs of linear motion and energy methods. Master these concepts to tackle complex rotational dynamics problems across engineering and physics.

Watch the full video summary here for detailed examples and derivations. Be sure to explore more chapters to strengthen your physics foundation.

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