Comprehensive Guide to Unit 5: Torque and Rotational Dynamics

Angular Position (θ)

The angle through which a point or line has rotated about an axis.

Standard Unit of Angular Position

Radians (rad).

Conversion of Degrees to Radians

360° = 2π rad.

Angular Velocity (ω)

The rate of change of angular position.

Formula for Angular Velocity

ω_avg = Δθ/Δt.

Unit of Angular Velocity

Radians per second (rad/s).

Direction Convention for Angular Velocity

Counter-clockwise (CCW) is positive, clockwise (CW) is negative.

Angular Acceleration (α)

The rate of change of angular velocity.

Formula for Angular Acceleration

α_avg = Δω/Δt.

Unit of Angular Acceleration

Radians per second squared (rad/s²).

Arc Length (s)

The linear distance traveled along the curve, s = rθ.

Tangential Velocity (v_t)

Instantaneous linear speed tangent to the path, v_t = rω.

Tangential Acceleration (a_t)

Linear acceleration tangent to the path, a_t = rα.

Kinematic Equations for Rotation

Equations that apply only when α is constant.

First Kinematic Equation for Rotation

ωf = ωi + αt.

Second Kinematic Equation for Rotation

Δθ = ω_i t + (1/2)α t².

Third Kinematic Equation for Rotation

ωf² = ωi² + 2α(Δθ).

Torque (τ)

The ability of a force to cause angular acceleration.

Torque Formula

τ = rFsinθ.

Factors that Affect Torque

How hard you push, where you push, and the angle at which you push.

Maximum Torque Condition

Occurs when the force is applied perpendicular to the lever arm.

Lever arm Distance (r)

Distance from the axis of rotation to where force is applied.

Static Equilibrium Condition 1

Translational Equilibrium: The net force is zero.

Static Equilibrium Condition 2

Rotational Equilibrium: The net torque is zero.

Rotational Inertia (Moment of Inertia, I)

Resistance to changes in angular velocity.

Rotational Inertia Formula

I = Σmr².

Factors Influencing Rotational Inertia

Total mass and mass distribution relative to the axis of rotation.

Moment of Inertia for a Hoop

I = MR² (All mass is far from center).

Moment of Inertia for a Solid Cylinder/Disk

I = (1/2)MR².

Moment of Inertia for a Solid Sphere

I = (2/5)MR².

Newton's Second Law for Rotation

Στ = Iα.

Rotational Kinetic Energy (K_rot)

K_rot = (1/2)Iω².

Total Kinetic Energy in Rolling Motion

Ktotal = Ktrans + K_rot.

Bridge Equation for Rolling Objects

v = Rω.

Conservation of Angular Momentum

If net external torque is zero, Li = Lf.

Angular Momentum (L)

L = Iω.

Angular Impulse Relationship

ΔL = τ_net Δt.

Common Mistake: Radians vs. Degrees

Use radians in kinematic formulas and degrees in geometry.

Common Mistake: Mixing v and ω

Do not confuse linear speed with angular speed.

Common Mistake: Pivot Point in Equilibrium

Torque must be calculated relative to a single axis of rotation.

Common Mistake: Ignoring Kinetic Energy Terms

Remember to include (1/2)Iω² in conservation of energy problems.

Common Mistake: Signs of Torque

Be consistent with CCW as positive and CW as negative.

Angular Momentum for Point Particles

L = mvr for point particles relative to an axis.