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.

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

• Kinetic Energy (KE): The energy an object possesses due to its motion.

• Formula: $KE = \frac{1}{2}mv^2$

• Work-Energy Theorem: The net work done on an object is equal to the change in its kinetic energy.

• Formula: $W_{net} = \Delta KE = KE_f - KE_i$

• Variable Forces: For non-constant forces, work is calculated as $W = \int_{x_i}^{x_f} F(x) dx$

• Power: The rate at which work is done or energy is transferred.

• Formula: $P = \frac{dW}{dt}$

• Example: Lifting a box vertically with a constant force does work equal to the change in gravitational potential energy.

Ch 7: Potential Energy and Energy Conservation

Potential Energy and Conservative Forces

Potential energy is the stored energy of position possessed by an object. Conservative forces, such as gravity and elastic (spring) forces, allow the definition of potential energy functions.

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

• Elastic Potential Energy (Ue): $U_e = \frac{1}{2}kx^2$ (for a spring)

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

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

Conservation of Mechanical Energy

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

• Mechanical Energy Conservation: $E_{mech} = KE + PE = \text{constant}$

• With Nonconservative Forces: $KE_i + PE_i + W_{nc} = KE_f + PE_f$

• Application: Calculating the speed of a roller coaster at different points using energy conservation.

• Special Case: When kinetic friction is present, mechanical energy is not conserved; energy is transformed into heat.

Ch 8: Linear Momentum and Collisions

Linear Momentum and Its Conservation

Linear momentum is a measure of an object's motion, and its conservation is a fundamental principle in isolated systems.

• Linear Momentum (p): $\vec{p} = m\vec{v}$

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

• Conservation of Momentum: In the absence of external forces, the total linear momentum of a system remains constant.

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

• Center of Mass: The point representing the average position of the mass in a system.

• Formula: $\vec{R}_{cm} = \frac{1}{M} \sum m_i \vec{r}_i$

Collisions: Elastic, Inelastic, and Completely Inelastic

Collisions are classified based on whether kinetic energy is conserved.

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

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

• Completely Inelastic Collision: Colliding objects stick together after the collision.

• 1D Collisions: Conservation laws are applied along a single axis.

• Example: Two carts colliding on a frictionless track.

Ch 9: Rotation of Rigid Bodies

Rotational Kinematics and Dynamics

Rotational motion describes the movement of rigid bodies around a fixed axis, analogous to linear motion but with angular quantities.

• Angular Position (θ): The angle describing the orientation of a line with respect to a reference direction.

• Angular Velocity (ω): $\omega = \frac{d\theta}{dt}$

• Angular Acceleration (α): $\alpha = \frac{d\omega}{dt}$

• Rotational Kinematic Equations: For constant angular acceleration:

  • $\omega_f = \omega_i + \alpha t$

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

  • $\omega_f^2 = \omega_i^2 + 2\alpha \theta$

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

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.

  • $I = \sum m_i r_i^2$ (discrete masses)

  • $I = \int r^2 dm$ (continuous mass distribution)

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

• Example: Calculating the moment of inertia for a solid disk about its center.

Summary Table: Types of Collisions

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?	Example
Elastic	Yes	Yes	Billiard balls colliding
Inelastic	Yes	No	Car crash with deformation
Completely Inelastic	Yes	No	Clay balls sticking together