Work, Energy, Momentum, and Rotation: Study Notes

Ch 6: Work and Kinetic Energy

Work, Kinetic Energy, and the Work-Energy Theorem

The concept of work and energy is fundamental in physics, describing how forces cause changes in motion and energy states. The work-energy theorem connects the work done by forces to the change in kinetic energy of a system.

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

• Constant and Variable Force: For a constant force, $W = Fd \cos\theta$. For a variable force, $W = \int_{x_i}^{x_f} F(x) dx$.

• Kinetic Energy (KE): The energy of motion, given by $KE = \frac{1}{2}mv^2$.

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

• Work with Varying Forces: When force varies with position, integrate the force over the path.

• Power (P): The rate at which work is done: $P = \frac{dW}{dt}$ or $P = Fv$ for constant force and velocity.

Example: A 2 kg object accelerated from rest to 3 m/s. The work done is $\Delta KE = \frac{1}{2} \times 2 \times (3^2 - 0^2) = 9$ J.

Ch 7: Potential Energy and Energy Conservation

Potential Energy and Conservation Laws

Potential energy is stored energy due to an object's position or configuration. Conservation laws describe how energy is transformed but not created or destroyed in isolated systems.

• Gravitational Potential Energy (Ug): $U_g = mgh$ (near Earth's surface).

• Elastic Potential Energy: For a spring, $U_{el} = \frac{1}{2}kx^2$.

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

• Nonconservative Forces: Forces like friction, where work depends on the path and mechanical energy is not conserved.

• Conservation of Mechanical Energy: In the absence of nonconservative forces, $E_{mech} = KE + PE$ is constant.

• Mechanical Energy with Friction: When friction is present, $E_{mech}$ decreases as energy is transformed into heat.

• Application: Energy conservation is used to solve problems involving motion, especially with circular or oscillatory motion.

Example: A mass slides down a frictionless hill: $mgh = \frac{1}{2}mv^2$ at the bottom.

Ch 8: Linear Momentum and Collisions

Momentum, Conservation, and Collisions

Momentum is a measure of an object's motion, and its conservation is a key principle in analyzing collisions and interactions.

• Linear Momentum (p): $p = mv$ for a particle; for a system, sum over all particles.

• Conservation of Linear Momentum: In an isolated system, total momentum remains constant: $\sum p_{initial} = \sum p_{final}$.

• Collisions in 1D: Analyze using conservation of momentum and, for elastic collisions, conservation of kinetic energy.

• Types of Collisions:

  • Elastic: Both momentum and kinetic energy are conserved.

  • Inelastic: Momentum conserved, kinetic energy not conserved.

  • Completely Inelastic: Colliding objects stick together after collision.

• Center of Mass (CM): The point representing the average position of mass in a system: $x_{CM} = \frac{\sum m_ix_i}{\sum m_i}$.

Example: Two carts of mass 2 kg and 3 kg moving towards each other at 1 m/s and -2 m/s, respectively. Total momentum before collision: $2 \times 1 + 3 \times (-2) = 2 - 6 = -4$ kg·m/s.

Ch 9: Rotation of Rigid Bodies

Rotational Kinematics and Dynamics

Rotational motion involves objects spinning about an axis. The principles of linear motion extend to rotation with analogous quantities.

• Angular Position (θ), Velocity (ω), and Acceleration (α): Describe rotational motion. $\omega = \frac{d\theta}{dt}$, $\alpha = \frac{d\omega}{dt}$.

• Rotation with Constant Angular Acceleration: Equations similar to linear kinematics apply:

  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$

  • $\omega = \omega_0 + \alpha t$

• Relating Linear and Angular Quantities: $v = r\omega$, $a = r\alpha$.

• Rotational Kinetic Energy: $KE_{rot} = \frac{1}{2}I\omega^2$, where $I$ is the moment of inertia.

• Moment of Inertia (I): Measures resistance to rotational acceleration: $I = \sum m_ir_i^2$ for discrete masses.

• Parallel Axis Theorem: $I = I_{CM} + Md^2$, where $d$ is the distance from the center of mass axis.

• Calculations of Moment of Inertia: Use geometry and mass distribution to determine $I$ for various shapes.

Example: For a solid disk of mass $M$ and radius $R$, $I = \frac{1}{2}MR^2$ about its central axis.