Study Notes on Rotational Motion and Angular Momentum
INTRODUCTION TO ROTATIONAL MOTION AND ANGULAR MOMENTUM
10.1 Angular Acceleration
Learning Objectives:
Describe uniform circular motion.
Explain non-uniform circular motion.
Calculate angular acceleration of an object.
Observe the link between linear and angular acceleration.
Uniform Circular Motion
Defined as motion in a circle at constant speed and constant angular velocity. The direction of the linear velocity vector (v) is always tangent to the circular path, while the angular velocity vector (\vec{\omega}) points along the axis of rotation, determined by the right-hand rule.
Angular velocity (\omega) defined as the time rate of change of angle (\frac{d\theta}{dt}). It is a vector quantity, with its magnitude often referred to as angular speed.
Relationship between angular velocity (\omega) and linear velocity (v) of a point at radius r is expressed as:
v = r \omega
Where r is the radius of the circular path. This equation shows that points further from the axis of rotation move with greater linear speeds for the same angular velocity.
Sign convention:
Counter clockwise direction is positive.
Clockwise direction is negative.
Non-uniform Circular Motion
Cases when angular velocity is not constant, leading to angular acceleration (\alpha). In non-uniform circular motion, a particle experiences both tangential acceleration (at = r\alpha) which changes its speed, and centripetal acceleration (ac = v^2/r = r\omega^2) which changes its direction. The total linear acceleration is the vector sum of these two components.
Angular acceleration (\alpha) defined as the rate of change of angular velocity:
\alpha = \frac{\Delta\omega}{\Delta t}
Units of angular acceleration are: radians per second squared (rad/s²).
Positive \alpha indicates increasing angular velocity; negative \alpha indicates decreasing angular velocity.
Example 10.1 - Angular Acceleration of a Bike Wheel
Scenario: A bike wheel accelerates from rest to a final angular velocity of 250 rpm in 5.00 s.
Part (a) - Calculate the angular acceleration (\alpha).
Strategy: Use the definition of angular acceleration:
\alpha = \frac{\Delta\omega}{\Delta t}
Convert final angular velocity from rpm to rad/s.
Part (b) - Calculate time taken to stop if brakes are applied causing angular acceleration of -20 rad/s².
Use the definition of angular acceleration to solve for time.
Discussion
In part (a), the angular acceleration is small and positive.
In part (b), larger negative angular acceleration leads to rapid stopping of the wheel.
Comparison with linear motion implies similar dynamics in angular motion.
10.2 Kinematics of Rotational Motion
Learning Objectives:
Observe the kinematics of rotational motion.
Derive rotational kinematic equations.
Evaluate problem-solving strategies for rotational kinematics.
Kinematic Relationships
Kinematics describes the relationships among rotation angle (\theta), angular velocity (\omega), angular acceleration (\alpha), and time.
A familiar kinematic equation for translational motion is:
d = v_0 t + \frac{1}{2} a t^2
For rotational motion, the equivalent relationship is:
\theta = \theta0 + \omega0 t + \frac{1}{2} \alpha t^2
Angular acceleration (\alpha) is constant.
Relationships for rotational and translational kinematics can be summarized in Table 10.2.
Example 10.3 - Acceleration of a Fishing Reel
Context: A deep-sea fisherman reels in fish, causing the line to unwind from a reel with angular acceleration of 4.00 rad/s².
Calculate:
(a) Final angular velocity after 2.00 s.
(b) Speed of line leaving reel.
(c) Number of revolutions made.
(d) Linear distance of fishing line that unwinds.
Strategy: Identify known quantities, select appropriate equations, and solve.
Problem-solving Strategy for Rotational Kinematics
Identify involvement of rotational kinematics.
Determine unknown quantities.
List known values.
Use appropriate equations.
Substitute known values with units to find numerical solutions.
Verify the result.
10.3 Dynamics of Rotational Motion: Rotational Inertia
Learning Objectives:
Understand the relationship between force, mass, and acceleration.
Study the turning effect of force (torque).
Explore analogies between force and torque, mass and moment of inertia.
Torque and Angular Acceleration
Torque (\tau) defined as the turning effectiveness of a force, determining how effectively a force causes or changes rotational motion. It depends on the magnitude of the force, the distance from the pivot (lever arm), and the angle at which the force is applied. Mathematically, it's expressed as: \vec{\tau} = \vec{r} \times \vec{F}, or in magnitude, \tau = r F \sin(\phi), where \phi is the angle between the position vector \vec{r} and the force vector \vec{F} .
Change in angular acceleration (\alpha) results from applying torque:
\tau = I \alpha
Where I is the moment of inertia, analogous to mass in linear motion. Moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on its mass and how that mass is distributed around the axis of rotation.
To calculate I for a point mass, consider:
I = m r^2
Where r is the perpendicular distance of the mass m to the axis of rotation. For a system of point masses, the total moment of inertia is the sum of individual moments: I = \Sigma mi ri^2. For extended objects, it involves integration.
Example 10.7 - Angular Acceleration on a Merry-Go-Round
Scenario: Father pushes a playground merry-go-round with a force of 250 N.
(a) Calculate the angular acceleration when no one is on.
(b) Calculate angular acceleration with an 18.0-kg child present.
Use torque definition and moments of inertia calculations to find results.
10.4 Rotational Kinetic Energy
Learning Objectives:
Derive equations for rotational work.
Calculate rotational kinetic energy.
Demonstrate the Law of Conservation of Energy related to rotational motion.
Rotational Work and Energy
Work is done when a force causes an object to rotate:
W = F \times d
where d is the arc length.
Torque's contribution to work leads to:
W = \tau \times \theta
Rotational kinetic energy (K) is defined as:
K = \frac{1}{2} I \omega^2
Example 10.8 - Work and Energy for Spinning a Grindstone
Situation: Calculation of work done and final angular velocity when spinning a grindstone.
(a) Calculate the work done with forces and angles provided.
(b) Analyze rotational kinetic energy using moment of inertia and angular velocity.
10.5 Angular Momentum and Its Conservation
Learning Objectives:
Understand the similarity between angular momentum and linear momentum.
Observe relationships of torque and angular momentum.
Apply conservation laws regarding angular momentum.
Definition of Angular Momentum
Angular momentum (L) defined as:
L = I \omega
Units: kg·m²/s.
Conservation of Angular Momentum
Angular momentum (L) is conserved in the absence of external torques. This means if the net external torque acting on a system is zero, its total angular momentum remains constant: \tau_{\text{net}} = \frac{dL}{dt} = 0 \implies L = \text{constant}.
This principle is expressed as: I1 \omega1 = I2 \omega2.
Exploring implications through examples with skaters, Earth, and rotating systems. For instance, an ice skater spinning faster by pulling her arms in reduces her moment of inertia (I) and consequently increases her angular velocity (\omega) to conserve angular momentum.
10.6 Collisions of Extended Bodies in Two Dimensions
Learning Objectives:
Explore the dynamics of collisions involving extended bodies.
Understand how angular momentum applies during interactions.
Collision Dynamics
Example discussed where a disk strikes a stick, illustrating angular momentum conservation.
Kinetic energy analysis highlights differences in elastic vs inelastic collisions.
10.7 Gyroscopic Effects
Learning Objectives:
Understand how to apply the right-hand rule.
Explain gyroscopic effects in rotating bodies like Earth.
Right-hand Rule
The right-hand rule determines direction of angular momentum and torque.
Gyroscopic precession explained using examples of spinning tops and