Work, Energy, and Momentum: Exam 3 Review Notes

Work, Energy, and Momentum

Dot Product

The dot product (also called the scalar product) is a mathematical operation that combines two vectors to produce a scalar quantity. It is fundamental in physics for calculating work and projections.

• Definition: The dot product of vectors \( \vec{A} \) and \( \vec{B} \) is given by:

$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

• Where \( \theta \) is the angle between the two vectors.

• Properties: Commutative (\( \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A} \)), distributive over addition.

• Application: Used to calculate work done by a force.

• Example: If a force of 10 N acts at 30° to the direction of motion over 5 m: $W = Fd\cos\theta = 10 \times 5 \times \cos 30^\circ = 43.3$ J.

Work

Work is the energy transferred to or from an object via the application of force along a displacement.

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

• Work is positive if the force has a component in the direction of displacement, negative if opposite.

• Units: Joule (J), where 1 J = 1 N·m.

• Example: Lifting a 2 kg object by 3 m: $W = mgh = 2 \times 9.8 \times 3 = 58.8$ J.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion.

• Formula: $K = \frac{1}{2}mv^2$

• Where m is mass and v is velocity.

• Units: Joule (J).

• Example: A 1 kg ball moving at 4 m/s: $K = \frac{1}{2} \times 1 \times 4^2 = 8$ J.

Energy Principle (Work-Energy Theorem)

The work-energy theorem relates the net work done on an object to its change in kinetic energy.

• Statement: $W_{\text{net}} = \Delta K = K_f - K_i$

• All forms of energy (kinetic, potential, etc.) can be included for a complete energy analysis.

• Example: If 20 J of work is done on a stationary object, its kinetic energy increases by 20 J.

Work Done by Different Forces

Different forces do work in different ways, depending on their nature and direction relative to displacement.

• Gravitational Force: $W_g = -mg\Delta y$ (negative if object is lifted against gravity)

• Normal Force: Usually does no work if perpendicular to displacement.

• Friction: $W_f = -f_k d$ (always opposes motion, so work is negative)

• Example: Sliding a box across a rough floor: friction does negative work, reducing kinetic energy.

Power

Power is the rate at which work is done or energy is transferred.

• Definition: $P = \frac{dW}{dt}$

• For constant force and velocity: $P = Fv\cos\theta$

• Units: Watt (W), where 1 W = 1 J/s.

• Example: Lifting a 10 kg mass at 2 m/s: $P = mgv = 10 \times 9.8 \times 2 = 196$ W.

Potential Energy

Potential energy is stored energy due to position or configuration.

• Gravitational Potential Energy: $U_g = mgh$

• Elastic (Spring) Potential Energy: $U_s = \frac{1}{2}kx^2$

• Where k is the spring constant, x is displacement from equilibrium.

• Example: Compressing a spring by 0.1 m with $k = 200$ N/m: $U_s = 0.5 \times 200 \times 0.1^2 = 1$ J.

Springs, Hook's Law, and Work Done by a Spring

Hook's Law describes the force exerted by a spring:

• Formula: $F_s = -kx$

• Negative sign indicates force is opposite to displacement.

• Work Done by a Spring: $W = -\Delta U_s = \frac{1}{2}k(x_i^2 - x_f^2)$

• Example: Stretching a spring from 0 to 0.2 m with $k = 50$ N/m: $W = 0.5 \times 50 \times (0.2^2 - 0^2) = 1$ J.

Momentum and Change of Momentum

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

• Formula: $\vec{p} = m\vec{v}$

• Change in Momentum: $\Delta \vec{p} = m\Delta \vec{v}$

• Total Momentum of a System: Sum of momenta of all objects: $\vec{p}_{\text{total}} = \sum m_i \vec{v}_i$

• Example: Two carts, 1 kg at 2 m/s and 2 kg at -1 m/s: $p_{\text{total}} = 1 \times 2 + 2 \times (-1) = 0$ kg·m/s.

Impulse and Its Relation to Momentum

Impulse is the product of average force and the time interval over which it acts; it equals the change in momentum.

• Formula: $\vec{J} = \vec{F}_{\text{avg}} \Delta t = \Delta \vec{p}$

• Units: kg·m/s or N·s.

• Example: A 5 N force acts for 0.2 s: $J = 5 \times 0.2 = 1$ N·s = 1 kg·m/s.

Conservation of Momentum

The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act.

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

• Applies to collisions and explosions.

• Example: Two ice skaters push off each other and move in opposite directions; their total momentum remains zero.

Elastic and Inelastic Collisions in 1D

Collisions are classified by whether kinetic energy is conserved.

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

• Inelastic Collision: Momentum is conserved, but kinetic energy is not. In a completely inelastic collision, objects stick together.

• Formulas:

For two objects, 1 and 2, with masses $m_1$, $m_2$ and velocities $v_1$, $v_2$:

Conservation of momentum: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

For elastic collisions, also: $\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2$

• Example: Two billiard balls collide elastically; both momentum and kinetic energy are conserved.

Explosions

In an explosion, a single object breaks into two or more pieces. The total momentum before and after the explosion is conserved (assuming no external forces).

• Application: Fireworks, rocket propulsion.

• Example: A stationary object explodes into two pieces of equal mass; they move in opposite directions with equal speed.