U12: Work and Energy

Conservation of energy is the principle that energy can neither be created nor destroyed

● Energy is the ability to do work

● Even as scientists discovered new forms of energy, conservation of energy has always been found to apply

The work done on a system by a constant force is defined as the product of the component of the force in the direction of motion times the distance through which the force acts

● 𝑊 = 𝐹𝑑 𝑐𝑜𝑠𝜃

○ F is the magnitude of force on the system

○ d is the magnitude of the displacement of the system

○ 𝜃 is the angle between the force vector and the displacement vector

● Work and energy are measured in newton-meters

Net work is the work done by the net external force 𝐹𝑛𝑒𝑡

● 𝑊net = 𝐹net𝑑 𝑐𝑜𝑠𝜃

○ If 𝜃 = 0(so cos = 1) then 𝑊net = 𝑚𝑎𝑑

■ This eventually leads to 𝑊 = .5𝑚𝑣2 − .5𝑚𝑣20

● This expression is called the work-energy theorem

Kinetic energy is the energy an object has by reason of its motion

● 𝐾𝐸 = .5𝑚𝑣2

Gravitational potential energy is the energy an object has due to its position in a gravitational field

● 𝑃𝐸g = 𝑚𝑔ℎ

○ This applies for any path that has a change in height

A conservative force is one in which work done by or against it depends only on the starting and ending points of a motion and not on the path taken

● Potential energy is the energy a system has due to position, shape, or configuration

○ It is stored energy that is completely recoverable

● The potential energy of a spring is the stored energy of a spring as a function of its displacement

○ 𝑃𝐸s = .5𝑘𝑥2

■ x is the distance the spring is compressed or extended

■ k is the spring constant

Conservation of mechanical energy is the rule that the sum of the kinetic energies and potential energies remains constant only if conservative forces act on and within a system

● 𝐾𝐸i + 𝑃𝐸i = 𝐾𝐸f + 𝑃𝐸f

● Mechanical energy is the total kinetic plus potential energy of a system (KE+PE)

A nonconservative force is one in which work depends on the path taken

● It adds or removes mechanical energy from the system

Efficiency is a measure of the effectiveness of the input of energy to do work

● 𝐸𝑓 𝑓 = 𝑊out/𝐸in

○ 𝑊out is the useful energy or work output

○ 𝐸in is the total energy input

Power is the rate at which work is done

● 𝑃 = 𝑊/𝑡

● The unit for power is the watt