Work

W = Fdcosθ

F is force, d is the distance over which the force is applied, and θ is the angle between the force and distance.

Units of Work

Joule = N·m = kg·m/s^{2}·m = kg·m^{2}/s^{2}

Sign Conventions of Work

If the force and the distance applied is in the same direction, work is positive.

If the force and the distance applied is in opposite directions, work is negative.

When is no work done if a force is acting in one direction?

If the force is acting in one direction, but the object moves in a perpendicular direction, then no work is done.

Work of Gravitational Force

Unlike friction, gravity always acts downwards. Thus, it does not matter what detour you take because sideward motion perpendicular to the gravitational force involves no work.

Does the amount of work depend on the path?

No, work is path independent. Pushing an object at constant speed up a frictionless inclined plane involves the same amount of work as directly lifting the same object to the same height at constant speed.

Sliding down a frictionless inclined plane involves the same gravitational work as doing a free fall at the same height.

Mechanical Advantage

Mechanical advantage = little input force (effort) -> large output force.

Work-kinetic energy theorem

Work on an object can transform into kinetic energy and kinetic energy can do work.

When you are pushing on an object, what is the expression for work?

Fd = ½mv^{2}

When gravity does work on an object, what is the expression for work?

F_{weight}h = mgh = ½mv^{2}

Power

Power is the rate of work, or work over time: P = W/t

The unit for power is the Watt, or W (don't confuse this W with the shorthand of work).

Watt = Joule / second

Kinetic Energy

½mv^{2}

Units of Kinetic Energy

Joule (kg m^{2}/s^{2})

Local Gravitational Potential Energy

mgh

Spring Potential Energy

(1/2)kx^{2}

Is the k value larger or smaller for stiffer springs?

Stiff springs have a larger k because they are harder to stretch (it takes more energy to stretch them).

General Gravitational Potential Energy

Conservation of Energy

The total amount of energy before = the total amount of energy after.

Conservative Forces

If a force doesn't dissipate heat, sound or light, then it is a conservative force.

Work done by conservative forces are path (dependent/independent)

Work done by conservative forces are path independent.

Non-Conservative Forces

non-conservative include frictional forces and human exertion. When friction acts on an object, the heat and sound released cannot be recovered. When you flex your arm, you lose heat that cannot be recovered (you cannot re-absorb the heat you lost).

Power Units

The unit for power is the Watt, or Joule per second.