Comprehensive Guide to Mechanics: Work, Energy, and Power

Work (W)

The process of transferring energy to or from a system by applying a force over a displacement.

Force

An interaction that changes the motion of an object; must be applied for work to be done.

Displacement

The distance an object moves in a specific direction; necessary for work to be done.

Positive Work (+W)

Occurs when the force aids motion (0° ≤ θ < 90°), adding energy to the system.

Negative Work (-W)

Occurs when the force opposes motion (90° < θ ≤ 180°), removing energy from the system.

Zero Work (0)

Occurs when the force is perpendicular to motion (θ = 90°), or there is no displacement.

Work Formula

W = F d cos(θ), where F is force, d is displacement, and θ is the angle between them.

Kinetic Energy (K)

The energy an object has due to its motion, calculated as K = 1/2 mv².

Work-Energy Theorem

States that the net work done on an object equals the change in its kinetic energy (W_net = ΔK).

Conservative Forces

Forces where work done is path-independent and energy can be fully recovered.

Non-Conservative Forces

Forces where work done depends on the path taken and energy is dissipated.

Gravitational Potential Energy (U_g)

The energy stored due to an object's height near the Earth's surface, U_g = mgh.

Elastic Potential Energy (U_s)

The energy stored in a compressed or stretched spring, U_s = 1/2 kx².

Mechanical Energy (ME)

The sum of kinetic and potential energies in a system, ME = K + Ug + Us.

Open System

A system where energy can enter or leave via external work (Ei + Wext = E_f).

Closed System

A system where no energy enters or leaves; total energy remains constant (Ei = Ef).

Power (P)

The rate at which work is done or energy is transferred, measured in Watts.

Average Power (P_avg)

Calculated as Work done divided by time interval, P_avg = Work/Δt.

Instantaneous Power (P)

Power at a specific instant, calculated as P = F v cos(θ) for constant velocity.

Cosine Function

A trigonometric function used in calculating work, representing the angle between force and displacement.

Force vs. Position Graph

A graph used to find the work done by a variable force by analyzing the area under the curve.

Energy Bar Charts (LOL Diagrams)

Visual tools to track energy flow, representing initial, flow, and final states of energy.

Internal Energy (ΔE_int)

Energy converted from mechanical energy into heat due to non-conservative forces like friction.

System Definition Trap

A common mistake in physics regarding if the Earth is included in the system when calculating energy.

Spring Constant (k)

A measure of a spring's stiffness, used in calculating elastic potential energy.

Height Reference (h)

The reference point chosen for potential energy calculations, typically set at the lowest point in problems.

Work done by Gravity

Depends on the system definition; can be internal or external based on whether the Earth is included.

Length of Spring in Energy Calculations

In U_s = 1/2 kx², x is the displacement from equilibrium, not the total length of the spring.

Kinetic Energy and Speed Relationship

Kinetic energy is proportional to the square of the speed; doubling speed quadruples kinetic energy.

Work done on an object

Needs a force applied, displacement, and a force component in the same direction for positive work.

Energy Conservation Laws

Principles that dictate how energy is conserved or transformed within defined systems.

Area under the Force vs. Position Curve

Represents the total work done by a variable force; can be negative if below the x-axis.

Angle θ in Work Calculations

The angle between the force vector and displacement vector; affects the calculation of work done.

Scalar Quantity

A quantity that has magnitude only, such as kinetic energy, without a direction.

Conservative Forces Examples

Gravity, spring force, electrostatic force; energy can be recovered.

Non-Conservative Forces Examples

Friction, air resistance; energy is dissipated and cannot be easily recovered.

Effect of Friction

Converts mechanical energy to internal energy (heat), reducing total mechanical energy in a system.

Height (h) in Potential Energy

Defined with respect to a reference level that you choose for calculating gravitational potential energy.

Friction's Work

Always negative when opposing displacement in mechanics.

Energy Transformation in Systems

Describes how energy changes forms, such as from potential to kinetic in examples like roller coasters.

Mechanical Energy Conservation Condition

Occurs when only conservative forces are acting, with no energy lost to heat or other forms.

Equilibrium Position in Springs

The unstretched position of a spring; used to measure displacement in potential energy equations.