Work, Energy, Momentum, and Rotation: Structured Study Notes

Ch 6: Work and Kinetic Energy

Work, Force, and Energy

Work is a measure of energy transfer that occurs when an object is moved by a force. The concept of work is fundamental in understanding how forces cause changes in energy.

• Work (W): Defined as the product of the force applied to an object and the displacement in the direction of the force.

• Formula: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

• Variable and Constant Forces: For variable forces, work is calculated using integration: $W = \int_{x_i}^{x_f} F(x) dx$

• Kinetic Energy (KE): The energy an object possesses due to its motion. $KE = \frac{1}{2}mv^2$

• Work-Energy Theorem: The net work done on an object is equal to the change in its kinetic energy: $W_{net} = \Delta KE$

• Power: The rate at which work is done. $P = \frac{dW}{dt}$

Example: If a 2 kg object is pushed with a constant force of 10 N over a distance of 5 m, the work done is $W = 10 \times 5 = 50$ J.

Ch 7: Potential Energy and Energy Conservation

Types of Potential Energy

Potential energy is stored energy due to an object's position or configuration. It is a key concept in energy conservation.

• Gravitational Potential Energy: Energy due to an object's position in a gravitational field. $U_g = mgh$

• Elastic Potential Energy: Energy stored in elastic materials, such as springs. $U_e = \frac{1}{2}kx^2$

Conservative and Nonconservative Forces

• Conservative Forces: Forces for which the work done is independent of the path taken (e.g., gravity, spring force).

• Nonconservative Forces: Forces like friction, where work depends on the path and energy is dissipated as heat.

Conservation of Mechanical Energy

• Principle: In the absence of nonconservative forces, the total mechanical energy (kinetic + potential) of a system remains constant.

• Formula: $E_{mech} = KE + PE = \text{constant}$

• With Friction: When kinetic friction is present, mechanical energy decreases: $E_{mech, final} = E_{mech, initial} - W_{friction}$

Application of Energy Conservation

• Used to solve problems involving motion, especially when forces are difficult to analyze directly.

• Common in circular motion and systems with springs or gravity.

Example: A pendulum swings from a height; its potential energy converts to kinetic energy at the lowest point.

Ch 8: Linear Momentum and Collisions

Linear Momentum

Momentum is a measure of an object's motion, defined as the product of mass and velocity.

• Definition: $\vec{p} = m\vec{v}$

• System of Particles: Total momentum is the vector sum of individual momenta.

Conservation of Linear Momentum

• Law: In a closed system with no external forces, total linear momentum remains constant.

• Formula: $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$

Collisions

• Elastic Collisions: Both momentum and kinetic energy are conserved.

• Inelastic Collisions: Momentum is conserved, but kinetic energy is not.

• Completely Inelastic Collisions: Colliding objects stick together after impact.

Center of Mass

• Definition: The point where the total mass of a system can be considered to be concentrated.

• Formula (for discrete particles): $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

Example: Two ice skaters push off from each other; their combined momentum before and after is equal.

Ch 9: Rotation of Rigid Bodies

Angular Position, Velocity, and Acceleration

Rotational motion describes how objects spin around an axis. Key quantities include angular position, velocity, and acceleration.

• Angular Position ($\theta$): The angle an object has rotated, measured in radians.

• Angular Velocity ($\omega$): Rate of change of angular position. $\omega = \frac{d\theta}{dt}$

• Angular Acceleration ($\alpha$): Rate of change of angular velocity. $\alpha = \frac{d\omega}{dt}$

Rotation with Constant Angular Acceleration

• Analogous to linear kinematics, with equations for angular displacement, velocity, and acceleration.

• Equations:

  • $\omega_f = \omega_i + \alpha t$

  • $\theta = \omega_i t + \frac{1}{2}\alpha t^2$

Relating Linear and Angular Kinematics

• Relationship: $v = r\omega$ and $a = r\alpha$

• Where $r$ is the radius from the axis of rotation.

Rotational Kinetic Energy and Moment of Inertia

• Rotational Kinetic Energy: $KE_{rot} = \frac{1}{2}I\omega^2$

• Moment of Inertia (I): A measure of an object's resistance to changes in rotational motion. $I = \sum m_i r_i^2$

Parallel Axis Theorem

• Theorem: Used to find the moment of inertia about any axis parallel to one through the center of mass.

• Formula: $I = I_{cm} + Md^2$

• Where $I_{cm}$ is the moment of inertia about the center of mass, $M$ is total mass, and $d$ is the distance between axes.

Calculations of Moment of Inertia

• Depends on mass distribution and axis of rotation.

• Example: For a solid cylinder of mass $M$ and radius $R$, $I = \frac{1}{2}MR^2$

Collision Type	Momentum Conserved?	Kinetic Energy Conserved?
Elastic	Yes	Yes
Inelastic	Yes	No
Completely Inelastic	Yes	No

Additional info: Academic context and formulas have been expanded for clarity and completeness.