Phys101Lec17B

Page 1: Introduction to Rotational Motion

• Key topics:

  • Rotational Kinematics

  • Rotational Dynamics; Torque and Moment of Inertia

• References: 9-1, 2, 3, 4, 5

Page 2: Angular Quantities

• In rotational motion, points on the object move in circles around the axis of rotation (O).

• Radius (R) of the circles is significant; all points along a line through the axis move through the same angle in the same time.

• Angle (θ) in radians is defined as:

  • θ = l/R (where l is the arc length)

• Sign convention:

  • • for Counter-Clockwise (CCW)

  • • for Clockwise (CW)

• Relationship for one revolution:

  • l = 2πR (encodes 360° rotation)

Page 3: Angular Displacement and Velocity

• Angular Displacement (Δθ): Δθ = θ2 - θ1

• Average Angular Velocity (ω): ω = Δθ / Δt

• Instantaneous Angular Velocity: ω = lim (Δθ/Δt) as Δt approaches 0

Page 4: Frequency and Period

• Frequency (f) is the number of complete revolutions per second (unit: Hertz).

• Period (T) is the time taken for one complete revolution.

• Angular Velocity (ω in rad/sec): ω = 2πf (since there are 2π radians in a revolution).

Page 5: Angular Acceleration

• Average Angular Acceleration (α): α = (ω2 - ω1) / Δt

• Instantaneous Angular Acceleration: α = lim (dω/dt) as dt approaches 0

Page 6: Linear and Angular Velocities

• Points further from the axis of rotation move faster than those closer to the axis.

• Rotation relationship as angles measured in radians.

Page 7: Linear Accelerations

• Changing angular velocity leads to tangential acceleration:

  • Tangential acceleration (at) = αR (R is radius)

• Centripetal acceleration (

  • Centripetal acceleration (ac): ac = v^2/R

• Total linear acceleration (a): a = √(at^2 + ac^2) (vector sum)

Page 8: Example - Carousel

• A carousel has constant angular acceleration α = 0.060 rad/s² starting from rest for 8.0 s.

  • (a) Angular velocity (ω): ω = ω0 + αΔt = 0 + (0.060)(8.0) = 0.48 rad/s

  • (b) Linear velocity (v) of a child at R = 2.5 m: v = ωR = 0.48 × 2.5 = 1.2 m/s

  • (c) Tangential acceleration (at): at = αR = (0.060)(2.5) = 0.15 m/s²

  • (d) Centripetal acceleration (ac): ac = v²/R = (1.2²/2.5) = 0.58 m/s²

  • (e) Total linear acceleration (a): a = √(at² + ac²) = √(0.15² + 0.58²) = 0.60 m/s²

Page 9: Constant Angular Acceleration

• Equations of motion for constant angular acceleration parallel those for linear motion, substituting angular quantities in.

• Initial position is taken as zero.

Page 10: Example - Centrifuge Rotor

• Centrifuge accelerated from rest to 20,000 rpm in 30 s.

  • (a) Average angular acceleration (α): α = Δω / Δt = (20000 RPM to rad/s: 20000 × 2π/60) / 30

  • (b) Revolutions made (θ) during acceleration:

    • θ = ω0t + 0.5αt², using Δω and Δt for calculation.

Page 11: Rotational Dynamics

• Importance of long lever arms in rotating objects: a force is needed to initiate rotation.

Page 12: Torque

• Torque (τ) is defined as the force applied at a distance (lever arm) from the axis of rotation:

  • Lever arm = perpendicular distance from the axis to the force line.

Page 13: Lever Arms and Forces

• Example: Lever arms for different forces (FA, FD, FC) in a rotational system.

  • Importance of the location of force: Forces not shifted in free-body diagrams (FBD).

Page 14: Torque Calculation

• Torque formula:

  • τ = R F⊥ = RF sin(θ)

• Where F⊥ = F sin(θ) and R is the distance from the axis of rotation.

Page 15: Example - Torque on a Compound Wheel

• Two disk-shaped wheels (RA = 30 cm, RB = 50 cm) | Forces of 50 N acting on each.

  • Calculate net torque using τ = - (FR_A sin θ_A + FR_B sin θ_B)

Page 16: Dynamics of Rotation

• Tangential component of Newton’s second law applied to rotational systems.

• Equation: τ = Iα, where I = moment of inertia (I = mR²).

Page 17: Moment of Inertia

• Moment of inertia (I) = ∑ mR², an object's resistance to rotational motion.

• Distribution of mass affects resistance, even with same mass.

Page 18: Summary of Rotational Dynamics

• Summary of Newton’s Law for rotation: ∑τ = Iα

• Importance of mass distribution and axis of rotation in the calculation of rotational inertia.

Page 19: Table of Moments of Inertia

• Overview of common moments of inertia for various shapes:

  • Thin cylindrical shells, thin rods, solids, hollow cylinders, etc.

Page 20: Example - Atwood's Machine

• Atwood machine setup with masses mA and mB connected by a pulley with moment of inertia (I).

  • Derivation for acceleration of masses.

Page 21: Continuation of Atwood's Machine Analysis

• Forces acting on the masses and the relationship with torque and rotational motion derived from Newton's laws.