Work, Energy, Momentum, Rotational Dynamics, and Gravitation: Study Guide

Work and Energy

Work

Work is a measure of energy transfer when a force F acts on an object and causes displacement d. The angle θ between the force and displacement determines the effective component of force.

• Definition: Work (W) is given by $W = Fd \cos\theta$.

• Units: Joules (J)

• Key Point: Only the component of force in the direction of displacement does work.

• Example: Pushing a box along the floor with a force at an angle.

Mechanical Energy

Mechanical energy is the ability of an object to do work, and it exists in two primary forms: kinetic and potential energy.

• Kinetic Energy (KE): Energy due to motion. $KE = \frac{1}{2}mv^2$

• Potential Energy (PE): Energy due to position or configuration.

  • Gravitational Potential Energy: $PE_G = mgh$ (object at height h)

  • Spring Potential Energy: $PE_S = \frac{1}{2}kx^2$ (spring compressed/stretched by x)

• Conservative Forces: Potential energy can only be defined for conservative forces (e.g., gravity, spring force).

Work-Kinetic Energy Theorem

The work done by a force on an object equals the change in its kinetic energy.

• $W = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

• Application: Used to relate force, displacement, and velocity changes.

Conservation of Mechanical Energy

If only conservative forces are present, the total mechanical energy (sum of KE and PE) is conserved.

• $E = KE + PE$

• $E_i = E_f$ or $\Delta E = 0$

• $KE_i + PE_i = KE_f + PE_f$

• Application: Used to solve problems involving motion and energy changes.

Non-Conservative Forces

When non-conservative forces (e.g., friction) are present, mechanical energy is not conserved. The change in mechanical energy equals the work done by non-conservative forces.

• $\Delta E = W_{NC}$

• Example: Object compressing a spring and experiencing friction.

Power

Power is the rate at which work is done.

• Average Power: $\bar{P} = \frac{W}{\Delta t}$

• Constant Force: $\bar{P} = \vec{F} \cdot \vec{v}$

• Units: Watts (W)

Linear Momentum and Collisions

Linear Momentum

Linear momentum is a vector quantity representing the product of mass and velocity.

• $\vec{p} = m\vec{v}$

• Conservation: In the absence of external forces, total momentum is conserved.

Impulse

Impulse is the change in momentum resulting from a force applied over a time interval.

• $\vec{I} = \vec{F} \Delta t$

• $\vec{I} = \Delta \vec{p}$

• Generalized Newton's Second Law: $\vec{F} = \frac{\Delta \vec{p}}{\Delta t}$

Collisions

During collisions, total linear momentum is conserved. Kinetic energy may or may not be conserved, leading to different types of collisions.

• Elastic Collision: $KE_{Ti} = KE_{Tf}$

• Inelastic Collision: $KE_{Ti} > KE_{Tf}$

• Explosive Collision: $KE_{Ti} < KE_{Tf}$

• Momentum Conservation: $\vec{p}_{Ti} = \vec{p}_{Tf}$

Rigid Body and Center of Mass

Rigid Body

A rigid body is an object with a fixed shape that cannot be deformed. The center of mass is a point representing the average position of mass distribution.

• Center of Mass Coordinates:

  • $x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$

  • $y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$

• Application: Used to analyze motion and distribution of mass in systems.

Rotational Dynamics

Angular Quantities

Rotational motion involves angular displacement, velocity, and acceleration.

• Angular Displacement: $\Delta \theta = \theta_B - \theta_A$

• Angular Velocity: $\bar{\omega} = \frac{\Delta \theta}{\Delta t}$

• Angular Acceleration: $\bar{\alpha} = \frac{\Delta \omega}{\Delta t}$

• Linear-Angular Relations:

  • $v_T = \omega R$

  • $a_T = \alpha R$

  • $a_r = \frac{v^2}{R}$

Moment of Inertia

Moment of inertia quantifies the rotational mass distribution relative to an axis.

• $I = \sum_i m_i r_i^2$

• Depends on: Mass, axis location, mass distribution.

• Application: Used in calculating rotational kinetic energy and dynamics.

Rotational Kinetic Energy

Energy due to rotation is given by:

• $KE_{rot} = \frac{1}{2}I\omega^2$

• Rolling Object: $KE = KE_{trans} + KE_{rot} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$

Torque

Torque is the rotational equivalent of force, causing angular acceleration.

• $\tau = Fr \sin\theta$

• Direction: Counterclockwise ($\tau > 0$), Clockwise ($\tau < 0$)

• Newton's Second Law for Rotation: $\tau = I\alpha$

Angular Momentum

Angular momentum is a measure of rotational motion.

• $L = r m v \sin\theta$

• Rigid Body: $L = I\omega$

• Direction: Counterclockwise ($L > 0$), Clockwise ($L < 0$)

Conservation of Angular Momentum

If net external torque is zero, angular momentum is conserved.

• $\tau = \frac{\Delta L}{\Delta t}$

• If $\tau = 0$, $\Delta L = 0$ so $L_i = L_f$

• Example: Figure skater spinning faster by pulling arms in (reducing moment of inertia).

Universal Theory of Gravitation

Kepler's Laws

Kepler's laws describe planetary motion:

• 1. Planets move in elliptical orbits with the Sun at one focus.

• 2. Equal areas are swept in equal times: $\frac{\Delta A}{\Delta t} = \text{constant}$

• 3. Square of orbital period is proportional to cube of semi-major axis: $T^2 = K a^3$

Newton's Law of Universal Gravitation

Newton's law states that every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them.

• $F = G \frac{m_1 m_2}{r^2}$

• $G = 6.67 \times 10^{-11} \text{Nm}^2/\text{kg}^2$

Gravitational Potential Energy

The energy of a system of two masses separated by distance r is:

• $U(r) = -G \frac{m_1 m_2}{r}$

Total Energy in Gravitational Field

The total energy of an object of mass m moving in the gravitational field of mass M is:

• $E = \frac{1}{2}mv^2 - G\frac{mM}{r}$

• For circular orbits: $E = -\frac{1}{2}G\frac{mM}{r}$