Unit 6: Rotational Motion

Unit 6: Rotational Motion

Introduction

• Definition of Rotational Motion: Rotational motion is essential in everyday life. It includes the rotation of
    the Earth, wheels, and various technological applications, from Swiss watches to heavy machinery.

• Key Concepts: To understand rotational motion, three key concepts are crucial:
    - Angular Velocity
    - Angular Acceleration
    - Centripetal Acceleration

• Applications: These concepts help in understanding various motions:
    - A car on a circular racetrack
    - Clusters of galaxies orbiting a center

• Integration with Newton's Laws:
    - Rotational motion, combined with Newton’s law of universal gravitation, helps explain space travel and satellite placement.
    - Energy conservation principles and gravitational potential energy relate to planetary escape speed.

6.1 Rotation About a Fixed Axis

• Axis of Rotation: The axis about which an object rotates. Every rotating body has this axis.
    - Example: Earth has two axes of rotation.

Notes of Importance (N.B.):

• All parameters such as angular displacement (q), work (w), angular acceleration (a), time (t), and angular momentum (L) are represented using the right-hand rule of rotation about the axis O.

Angular Displacement

• Definition: The angular position of a rotating disc about a fixed axis.
• Reference Line: For the disc, a fixed line is chosen. A point P is located at a distance r from axis O.
• As the disc rotates, point P traces an arc length s on the circular path made by radius r.

Detailed Description of Angular Concepts

Angular Displacement Equation

• The relationship between angular displacement (ϴ), arc length (s), and radius (r):
  $\theta = \frac{s}{r}$
• SI Unit: The radian (rad) is defined as the angle subtended by an arc length equal to the radius of the arc,
  $1 rad = \frac{360°}{2\pi} = 57.3°$

Revolution to Radians Conversion

• 1 revolution = 360° = 2π radians

Angular Velocity (ω)

• Definition: The rate of change of angular displacement over time.
• Formula:
  $\omega = \frac{\Delta\theta}{t}$
• Measurement: Measured in radians/second (rad/s), it is a vector quantity.

Linear Velocity (V)

• Definition: The speed of a rotating object tangent to the curve along the circular path.
• Formula:
  $V = \omega \times r$
• Conversion: SI unit is meters/second (m/s) = rad/s × m.

Example Calculation:

• Calculate Angular Velocity of a Flywheel
    - Given: Radius = 2m, Linear velocity (V) = 16m/s.
    - Calculation:
  $\omega = \frac{V}{r} = \frac{16}{2} = 8 rad/s$

Exercises

1. Flywheel Spins: A flywheel spins 250 times per minute.
   What is its angular velocity?
      - Answer: 26.17 rad/s

2. CD Rotating: A CD rotates at 4800 revolutions per minute.
   What is the linear speed of a point at the rim (radius 40mm)?
      - Answer: 20.1 m/s

Worked Example 6.2

• Given: CD rotating at 4800 revolutions per minute.
• Find: Linear velocity of a point 40 mm from the axis of rotation.

1. Calculate Angular Velocity:
   $\text{Angle turned in 60 s} = 4800 \times 2\pi$
      - Use the equation:
   $\omega = \frac{\text{angle}}{t}$
2. Substituting Values:
   $\omega = \frac{4800 \times 2\pi}{60} = 502.7 rad/s$
3. Final Velocity Calculation:
   $v = r\omega = 0.04 m \times 502.7 rad/s = 20.1 m/s$

Review Questions

1. Car Engine: A car engine rotates at 3000 revolutions per minute.
   What is its angular velocity in rad/s?

2. Linear Velocity on a CD: The linear velocity of a point on a CD is a constant 1.2 m/s.
      - At the start (23 mm from center),
      - At the end (58 mm from center).
   What is the angular velocity?

3. Car Wheel: A car traveling at 16 m/s has a wheel with a diameter of 0.7 m.
   Find the wheel's angular velocity in rad/s.

4. Turbine: A turbine rotates at 3000 revolutions per minute.
      - a) What is its angular velocity in rad/s?
      - b) Calculate the linear velocity of a turbine blade at the end and halfway down the blade.

Angular Acceleration (α)

• Definition: Angular acceleration is the rate of change of angular velocity over time.
• Formula:
  $\alpha = \frac{\Delta\omega}{t}$
• SI Unit: rad/s²
• Other unit examples include rev/min².

Example of Angular Acceleration Calculation

• A flywheel accelerates from 15.1 rad/s to 23.1 rad/s over 4 seconds.

1. Angular Displacement (θ):
   $\theta = \left(\omega_{i} + \omega_{f}\right) \times \frac{t}{2} = \left(15.1 + 23.1\right) \frac{4}{2} = 76.4 rad$
2. Find α:
   $\alpha = \frac{\Delta\omega}{t} = \frac{23.1 - 15.1}{4} = 2 rad/s²$

Exercises

1. Calculate Angular Acceleration:
      - If angular velocity increases from 3 rad/s to 23 rad/s in 4 seconds, find α.
2. Final Angular Velocity: A flywheel starts from rest and rotates at 3 rad/s² for 5 seconds. Find the final angular velocity.

Rotational Kinematics

Constant Angular Acceleration

• The simplest motion to analyze is rotational motion under constant angular acceleration.

Exercises

1. Exercise Bike: Angular velocity at the rear wheel is 4 rad/s at t=0 with an acceleration of 2 rad/s².
      - a. Angle with x-axis at t=3 s?
      - b. Angular velocity at this time?

2. Disk Comparison: A disk rotates; Point B is three times farther from the axis than Point A. If the speed of B is V, what is the speed of A?

3. Disk with Increasing Angular Velocity: A disk's Point B is twice as far from the axis as Point A. Analyze the angular and radial acceleration of A and B.

4. Discus Thrower: An angular acceleration of 50 rad/s² moves a discus in a circle of radius 0.8 m. Find radial and tangential components of acceleration.

5. Fairground Wheel: A wheel makes 1 rev every 8 s and decelerates at -0.1 rad/s². What is the initial angular velocity of someone located at 4 m from the axis?

6.2 Torque and Angular Acceleration

• Torque (τ): The rotational effect of force. Defined and measured as:
  $\tau = r imes F \sin(\theta)$
    where
  $\theta$ is the angle between force and position vector.
• Units: Torque uses the unit N.m, which differs from Joules (work and energy).
  

Definition of Torque

• Torque can also be defined as:
  $\tau = r x F$

Vector Product of Two Vectors

• The vector product from unit lessons states that:
  $\tau = r x F = |r||F| \sin(\theta)$
• Remember in vector products, the orientation of forces plays a crucial role in determining the resultant torque.

Exercises

1. Force Torque: If a force of (3i + 4j) N acts at (120i + 50j) cm from a pivot, determine the torque produced. (Find the angle between r and F.)
2. Mechanic Torque: A mechanic applies a 270 N force at 50 cm from a nut pivot. What is the torque produced, and how much work is done when rotating it through a half turn?
3. Force at Position Calculation: Calculate torque exerted by (2N, -5N) at (3m, 1m) from the pivot.

6.3 Rotational Kinetic Energy and Moment of Inertia (MOI)

• Definition of Rotational Inertia: The rotating body's resistance to change in motion, equivalent to mass in linear motion.
    - The moment of inertia (I) influences how hard it is to change the rotation of a body.
    - Torque relationship:
  $\tau = I\alpha$

Moment of Inertia for Point Mass and Rigid Bodies

• For a point mass:
  $I = mr^2$
    - Where m is mass and r is the distance from the axis of rotation.
• For a rigid body:
    - Sum of moment of inertia for point masses:
  $I = \sum m_{i}r_{i}^2$

Specific Shapes and their Moment of Inertia

• Examples:
    - Thin disk:
  $I = \frac{1}{2} MR^2$
    - Slender rod: Calculations vary based on where the axis passes through (center or end).

Rotational Kinetic Energy

• Rotational Equivalent of Kinetic Energy: In linear motion, the kinetic energy is given by the equation
  $E_K = \frac{1}{2} mv^2$
    - For rotation, we rephrase as:
  $E_{rot} = \frac{1}{2} I\omega^2$
    - This energy also depends on the position of the axis of rotation.

Exercises

1. Point Mass Energy: Calculate the rotational kinetic energy of a 2kg point mass moving at 5 rad/s, 0.5m from the axis.
2. Drum Rotation: Find the rotational kinetic energy of a drum with an I of 16000 kg.m² and an angular speed of 3 rad/s.
3. Torque and Work Done Calculation: A person pulls on a cord on a drum with a diameter of 32cm. Find torque, work done, and the angle the drum rotates.
4. Cylinder Acceleration: Analyze the acceleration of a cylinder when a string is pulled with a force of 16N.
5. Rolling Ball's Total Kinetic Energy: Determine the total kinetic energy of a 5-kg ball rolling at 4m/s with a radius of 10cm.
6. Inclined Plane Roll: Analyze the motion of a ring, solid cylinder, and sphere rolling down a slope from a height of 1m without slipping. Determine the order and time taken to reach the bottom.

6.4 Work and Power

• Work Done by Torque: Work done can be calculated by:
  $W = \tau \times q$
    - Torque has a directional component, with cos(0°) highlighting the effects of linear force components.
    - SI Units: Work (Joules) can be formulated through torque and angular displacement.

Power in Rotational Motion

• Rotational Power: The rate of work done in a rotational context is denoted as:
  $P = \frac{W}{t} = \tau \times \omega$
• Compare to the linear power format:
  $P = \frac{W}{t} = F \times S/t = F \times V$

Exercises

1. Mechanic's Power Development: If a mechanic applies a torque of 160 N.m over three-quarters of a turn in 2 seconds, find the power developed.

6.5 Parallel Axis Theorem

• Parallel Axis Theorem: The moment of inertia about a parallel axis is calculated by:
  $I_P = I_{CM} + M d^2$
    - Where M is mass of the rigi body, and d is the distance between the two axes.

Exercises

1. Moment of Inertia Calculation: For an 8 kg rigid body with a MOI of 60 kg.m², rotated about a parallel axis 0.6m from the CM, calculate the MOI about the new axis.
2. Uniform Solid Sphere: Verify moment of inertia about its surface parallel axis from the center calculation results: Show as (7/5)mR².
3. Moment of Inertia Comparisons: Compute MOI for solid cylinder, hollow cylinder, and solid sphere when their axes are at the edge compared to the center.

6.6 Angular Momentum and Angular Impulse

• Angular Momentum (L): Similar to linear momentum, but expressed as:
  $L = I\omega ext{ or }L = mvr$
    - For point mass:
  $L = mr^2 \omega = r \times m \times v$

Angular Impulse Defined

• Definition of Linear Impulse:
  $J = F_{net} imes ext{change in time} = \Delta p$
• Angular Impulse:
  $J = \tau_{net} \times \Delta t = \Delta L$
• Units of Angular Impulse are represented as Newton-meter-second [J] = N.m.s.

Exercises

1. Flywheel Angular Momentum Calculation: Given a flywheel rotating at 1000 rev/min with an MOI of 50 kg.m², determine the angular momentum.
2. Rotating Flywheel: Analyze the influence of a 2 kg mass on a flywheel’s angular speed after a duration of 2.5 s.

6.7 Conservation of Angular Momentum

• Law of Conservation of Angular Momentum: When the net torque acting on a rotating body is zero, its angular momentum remains constant:
  $\tau_{net} = 0 \implies L = \text{constant}$
• Applications: Skating and diving scenarios help to demonstrate this principle.

Exercises

1. Skater's Angular Momentum Change: A skater spins with an initial moment of inertia of 1.2 kg.m² at 5 rad/s. If she extends her arms, and spins at 1.8 rad/s, find her new MOI.

6.8 Center of Mass of a Rigid Body

• Center of Mass (C.M.): A rigid body balances freely when supported at its center of mass.
• Geometric Center: For uniform rigid bodies such as discs, the C.M. is at geometric center.
• Non-uniform Bodies: C.M. shifts closer to denser mass ends, such as in irregular shapes or significant mass distributions (e.g., Sun-Earth).

Exercises

1. Determine C.M.: Given various point masses, find the overall center of mass for the system.

Additional Solved Problems

1. Helicopter Rotor: A rotor turns at 3.20 × 10^2 revolutions per minute. Find its angular velocity in radians per second, arc lengths traced, and average angular velocity during increased throttle.
2. Angular Acceleration Calculation: If angular velocity reduces from 15 rad/s to 9 rad/s in 3 seconds, calculate average angular acceleration.
3. Acceleration Summary: Evaluate a wheel's rotation with a constant angular acceleration of 3.50 rad/s² from an initial velocity of 2 rad/s over 2 seconds, understanding total distance traveled in radians and revolutions.
4. Compact Disc Player: Analyze speed changes within compact disc technology, gauging speed variations depending on read head positions.
5. Race Car Dynamics: Analyze the uniform acceleration of a race car, gauge centripetal dynamics over a circular race track, while maintaining equations for kinematic continuity.

Conclusion

These notes provide a comprehensive framework for understanding rotational motion principles along with practical exercises, reinforcing critical concepts related to angular dynamics.