# Chapter 6: Work and Energy

• Energy is conserved; can’t be created / destroyed
• Kinetic Energy describes motion => mv^2
• Energy on earth originates from the sun
• Energy on earth is stored thermally and chemically
• Chemical energy is released by metabolism
• Energy is stored as potential energy in objects

### Conservation of Energy

Energy cannot be created or destroyed, but can be transformed other energies of another form.

• Dissipation of “heat” – Energy could be lost by transforming into heat

### Work

Work done is the dot product (scalar) of the force and the displacement times cos θ of the angle.

• W = F Δd cos θ = unit J (Joules)
• Work is only considered when the force applied is not perpendicular to the displacement
• In a uniform circular motion, the work done is always 0

#### Kinetic Energy

Work = Σ Force ( ΔDisplacement) = (mass)(acceleration)(Displacement)

﻿$W=m(a\Delta d)=m\frac{1}{2}(v_{2f}-v_{2i})=\frac{1}{2}mv_{2f}-\frac{1}{2}mv_{2i}\Rightarrow \Delta K=K_{f}-K_{i} giving: K=\frac{1}{2}mv_{2}$﻿

Constant Force ﻿$\Rightarrow$﻿ ﻿$W=(F\cos \Theta )\Delta d$﻿

Variable Force ﻿$W\approx F(\cos\Theta )\Delta d_{1}+F(\cos\Theta )\Delta d....$﻿

#### Hooke’s Law

﻿$W=\frac{1}{2}kX_{2}\Rightarrow with F_{x}(X)\Rightarrow (\frac{1}{2}kx)(x)$﻿

#### Gravitational Potential Energy

﻿$W_{gravity}=(F\cos\Theta )\Delta d$﻿ for ﻿$W_{gravity}=mg(h_{i}-h_{f})$﻿ with ﻿$|g|=9.8m/s_{2}$﻿

• Take into consideration with only the parallel force to displacement Take ﻿$U_{grav}=mgh\Rightarrow W_{grav} =-\Delta U_{grav}$﻿

For an object to fall, ﻿$\Delta U_{grav}$﻿ < 0 for that the final Energy < initial Energy

#### Elastic Potential Energy

﻿$W_{elastic}=\frac{1}{2}k x_{12}\frac{1}{2}kx_{f2}$﻿

Similar to the gravitational potential energy, this energy is stored depending on distance, x.

#### Conservative vs Non‐conservative Forces

1. A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions.
2. A force is conservative when it does no work on an object moving around a closed path, starting and finishing at the same point.

### Gravitational, Elastic Spring and Electric Force are Conservative

﻿$Work_{done}=Work_{Conservative}+Work_{Non-conservative}$﻿

• ﻿$Work_{Non-conservative}=\Delta K+\Delta U=(K_{f}-K_{i})+(U_{f}-U_{i})$﻿
• ﻿$W_{Nc}=(K_{f}+U_{f})-(K_{i}+U_{i})$﻿
• ﻿$E=K+U$﻿
• ﻿$W_{Nc}=E_{f}-E_{i}$﻿
• ﻿$W_{Nc}=0$﻿﻿$0=E_{f}-E_{i}$﻿﻿$E_{f}=E_{i}$﻿

### Power

Power - the rate of change of energy over time.

• P = Work / Time = ﻿$ΔW/Δt$﻿ = Change in energy / time
• Given Joule/ s = watt (W)
• Average Power = Force ( Average Velocity)
• Instantaneous Power = Power = Force (Velocity)