Energy & Work
Understanding energy transformations and the work-energy theorem
Work is the transfer of energy that occurs when a force acts on an object and causes it to move. Work is only done when the force has a component in the direction of motion.
Work equals force times displacement times cosine of angle
Key Points About Work:
- Work is a scalar quantity (has magnitude but no direction)
- Units: Joules (J) = Newton·meter (N·m)
- When θ = 0°, cos(θ) = 1, maximum work is done
- When θ = 90°, cos(θ) = 0, no work is done (force perpendicular to motion)
- When θ > 90°, work is negative (force opposes motion)
Work by Variable Forces
When force varies with position, work is the area under the force vs. displacement graph:
Work as integral of force over displacement
Kinetic energy is the energy of motion. Any object with mass that is moving has kinetic energy.
Kinetic energy equals one-half mass times velocity squared
Important: Kinetic energy depends on the square of velocity. Doubling the speed quadruples the kinetic energy!
The work-energy theorem states that the net work done on an object equals its change in kinetic energy.
Net work equals change in kinetic energy
Expanded form of work-energy theorem
This powerful theorem allows us to solve many problems without knowing the details of the forces or acceleration. We only need to know the initial and final velocities.
Gravitational Potential Energy
Energy stored due to an object's position in a gravitational field. We measure it relative to a reference level (usually ground level).
Gravitational potential energy
Where h is the height above the reference level. PE increases as height increases.
Elastic Potential Energy
Energy stored in elastic materials (springs, rubber bands) when they are stretched or compressed.
Elastic potential energy in a spring
Where k is the spring constant (stiffness) and x is the displacement from equilibrium position.
In a closed system with only conservative forces (like gravity and springs), the total mechanical energy remains constant.
Initial energy equals final energy
Conservation of mechanical energy
Conservative Forces
Forces for which the work done is path-independent. Examples: gravity, spring force. These forces can store and release energy.
Non-Conservative Forces
Forces for which work depends on the path taken. Examples: friction, air resistance. These forces convert mechanical energy to thermal energy.
Energy with Non-Conservative Forces
When non-conservative forces (like friction) are present, mechanical energy is not conserved:
Energy equation with non-conservative work
Where Wnc is the work done by non-conservative forces (usually negative for friction).
Power is the rate at which work is done or energy is transferred.
Power equals work divided by time
Power equals force times velocity
Units: Watts (W) = Joules/second (J/s). One horsepower (hp) = 746 W.