Work, Energy, and Momentum: Study Notes for College Physics

Work and the Dot Product

Definition and Properties of Work

Work is a fundamental concept in physics, describing the energy transferred by a force acting over a distance. The work done by a force can be positive, negative, or zero, depending on the direction of the force relative to the displacement.

• Work Formula: $W = \vec{F} \cdot \Delta \vec{r}$ (dot product of force and displacement)

• Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \theta$

• If $\theta > 90^\circ$, $\cos \theta$ is negative, so work is negative.

• The dot product is a scalar (not a vector).

• Component form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y$

Work by Constant Force

When a constant force acts along a straight path, the work is given by:

• $W = F \Delta r \cos \theta$

• Units: Joules (J), where $1\,J = 1\,N \cdot m$

• Work can be positive (force in direction of motion), zero (force perpendicular), or negative (force opposite to motion).

Work in Physical Scenarios

Work by Gravity and Other Forces

Consider lowering a book at constant speed:

• Gravity does positive work when the object moves downward.

• Hand does negative work (opposes motion).

• Total work on the book is zero (forces balance).

Work by Tension in Circular Motion

When a rock is twirled in a horizontal circle at constant speed, the tension in the string does zero work because the force is always perpendicular to the displacement.

• Work by tension: $W_{\text{Tension}} = T \Delta r \cos 90^\circ = 0$

• No energy is transferred by tension in uniform circular motion.

Work by Gravity Along a Path

The work done by gravity depends only on the vertical displacement, not the path taken.

• Work by gravity: $W_{\text{grav}} = F_{\text{grav}} \Delta r \cos \theta$

• For a 1 kg mass moving 0.5 m down: $W_{\text{grav}} = 10\,N \times 0.5\,m \times (-1) = -5\,J$

• Gravity is a conservative force: work is path-independent.

Conservative and Non-Conservative Forces

Conservative Forces

Forces for which the work done is independent of the path are called conservative forces. Examples include gravity, spring force, and electric force.

• Work by conservative force: $\Delta PE = -W_{\text{by force}}$

• Kinetic friction is not a conservative force.

Spring Forces and Hooke's Law

Ideal springs obey Hooke's Law, where the force is proportional to the displacement:

• $\vec{F} = -k \vec{x}$

• $k$ is the spring constant (N/m).

• Work to stretch a spring: $W = \int_0^{x_f} kx \, dx = \frac{1}{2} k x_f^2$

Work with Variable Forces

Work from Force vs. Position Graphs

When force varies with position, the total work is the area under the F(x) curve.

• Work: $W = \int_{x_0}^{x_f} F(x) \, dx$

• For step function: $W = \frac{1}{2} F_0 x_0 + F_0 x_0 = 1.5 F_0 x_0$

• For linear force: $F(x) = \frac{F_0}{x_f} x$

• Work: $W = \int_0^{x_f} \frac{F_0}{x_f} x \, dx = \frac{1}{2} F_0 x_f$

Kinetic Energy and the Work-Energy Theorem

Kinetic Energy

Kinetic energy is the energy of motion:

• $KE = \frac{1}{2} m v^2$

• Units: Joules (J)

• Always positive or zero

Work-Energy Theorem

The net work done on an object equals its change in kinetic energy:

• $W_{\text{net}} = \Delta KE$

• For point-like objects: $W_{\text{net}} = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$

Potential Energy and Conservation of Energy

Potential Energy

Potential energy is associated with conservative forces:

• Gravitational: $PE_{\text{grav}} = m g y$

• Spring: $PE_{\text{spring}} = \frac{1}{2} k x^2$

Conservation of Mechanical Energy

If only conservative forces act, the total mechanical energy (KE + PE) is conserved:

• $PE_f + KE_f = PE_i + KE_i$

• If non-conservative forces (like friction) are present: $PE_f + KE_f = PE_i + KE_i + W_{\text{non-conservative}}$

Power

Definition and Calculation

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

• $P = \frac{W}{\Delta t}$

• Units: Watts (W), $1\,W = 1\,J/s$

• For constant force: $P = F v$

Momentum and Conservation

Momentum

Momentum is a vector quantity defined as:

• $\vec{p} = m \vec{v}$

• Units: kg·m/s

• For a system: $\vec{p}_{\text{tot}} = \sum_i m_i \vec{v}_i$

Conservation of Momentum

If the net external force on a system is zero, total momentum is conserved:

• $\vec{p}_{\text{tot, final}} = \vec{p}_{\text{tot, initial}}$

• Internal forces cancel by Newton's Third Law.

Collisions

Collisions are classified as:

• Elastic: Total kinetic energy is conserved.

• Inelastic: Total kinetic energy is not conserved.

• Perfectly inelastic: Objects stick together; maximum loss of KE.

Energy in Physical Systems

Energy Graphs and Mechanical Energy

Energy graphs show how potential energy varies with position. The total energy is the sum of kinetic and potential energies.

• At any point: $E_{\text{tot}} = PE + KE$

• Maximum KE occurs where PE is minimum.

Motion on Frictionless Ramps

For objects sliding down frictionless ramps of equal height, the speed at the bottom is the same regardless of the ramp's shape.

• Conservation of energy: $PE_i + KE_i = PE_f + KE_f$

• All start with same PE, end with same KE.

• With friction, some energy is lost as heat; final speed is less.

Summary Table: Conservative vs Non-Conservative Forces

Force Type	Path Dependence	Examples
Conservative	No	Gravity, Spring, Electric
Non-Conservative	Yes	Kinetic Friction, Air Resistance

Summary Table: Energy Formulas

Energy Type	Formula
Kinetic Energy	$KE = \frac{1}{2} m v^2$
Gravitational Potential Energy	$PE_{\text{grav}} = m g y$
Spring Potential Energy	$PE_{\text{spring}} = \frac{1}{2} k x^2$

Additional info: Academic context and explanations have been expanded for clarity and completeness. Only images directly relevant to the explanation have been included.