## I. 5.1: WORK

The product of the force (only the appropriate component) and distance

### A. W = Fd → W = Fdcosθ

1. θ is the angle between F and d

2. If F and d are parallel, cosθ = 1

### B. Work unit is Joule (N·m = kgm2/s2)

### C. Work is scalar

### D. EX:

Box with mass m (20 kg) is pushed 6 meters with a force F (100 N); μk = 0.4

1. How much work is done by F?

a) W = Fdcosθ = 100 N * 6 M * 1 = 600 J

2. How much work is done by the normal force?

a) F_{N} = F_{g}cosθ = m*g*cos(90) = 20 kg * 10 m/s^{2} * 0 = 0 N

3. How much work is done by gravity?

a) Since the gravitational force is perpendicular to the floor, the θ = 90, and cosθ = 0. Therefore, work done by gravity is 0.

4. How much work is done by the force of friction?

a) F_{F} = μ_{k }F_{N}; W = μ_{k }*mg*d*cosθ = 0.4*20kg*10m/s^{2}*1 = 80 N

b) W = Fdcosθ = 80N*6*cos180 = 80*6*(-1) = -480 J

5. What is the total work done on the crate?

a) Add all the work done by each of the forces

b) W_{tot} = W_{by F} + W_{by Fn} + W_{by Fg} + W_{by Ff }= 600 J + 0 + 0 + (-480) = 120J

### E. Affects of angle of force on Work

### F. *The same concept can be applied to work related to kinetic energy; see below

## II. 5.2: POWER

Measures how fast work gets done(in J/s)

### A. P = W/t

1. Power is measured in Watts (J/s)

2. Horsepower = 750 watts

### B. P = F*d/t = F*v

## III. 5.3: KINETIC ENERGY

Energy in objects that are moving and can exert a force; kinetic energy is the energy an object has after a force has been applied; the KE can perform work → KE units are the same as work!!

### A. To figure out how much kinetic energy an object has, use:

*v ^{2} = v_{0}^{2} + 2 ad*

### B. v^{2} = v_{0}^{2} + 2ad → v^{2} = 2ad → v^{2} = 2(F/m)d → v = √(2Fd/m)

### C. Rewrite → F*d = ½mv^{2} → W = ½mv^{2}

### D. F*d = Kinetic energy, so KE = ½mv^{2}

### E. WORK-ENERGY THEOREM

1. Above, we assumed that v_{0} = 0; what if it was not?

a) v^{2} – v_{0}^{2} = 2(F/m)d → ½m(v^{2} – v_{0}^{2}) = Fd → ½mv^{2} -½m v_{0}^{2} = Fd

b) → **W = ½mv ^{2} – ½mv_{0}^{2} = KE_{final} – KE_{initia}**

_{l}

c) ∴ **W _{tot} = ΔKE**

d) (+) W will add to KE, (-)W will take away from KE

## IV. 5.4: POTENTIAL ENERGY

The energy an object has by virtue of its position (gravitational, electrical, elastic)

### A. Gravitational → if a brick is lifted and placed on a shelf, how much did its potential energy change?

1. Gravity did work on the brick while the brick was lifted (negative work)

2. **W _{by Fgrav} = F_{grav}(-h) = -mgh**

3. ΔPE_{grav} = -W_{by Fgrav} =** –**mgh

### B. Gravity Is a Conservative Force

1. Compare a brick being lifted 2 m or being pushed up a ramp to the same height; the work done by gravity is the same

a) Large force over small distance vs small force over large distance

b) Work depends only on the initial and final positions, not the pathway!

### C. Friction Is Not a Conservative Force

1. Friction does depend on path taken

2. There is no such thing as “frictional potential energy”

### D. Springs – elastic potential energy

1. Hooke’s law – the amount of energy required to stretch/compress something elastic (like a spring) is directly proportional to the distance it will stretch

2. k = spring constant

3. F = -kx → x = distance

4. **W _{spring} = ½kx^{2}**

## V. 5.5: TOTAL MECHANICAL ENERGY

### Sum of kinetic energy and potential energy

### A. E = KE + PE = ½mv^{2} + mgh

### B. Conservation of Total Mechanical Energy

If the only forces acting on an object are conservative (e.g. no friction), then the object’s total mechanical energy will remain constant

1. Any 2 positions or times should have the same total mechanical energy

2. E_{i}* =* E_{f}

3. KE_{i} + PE_{i} = KE_{f} + PE_{f}

### C. Using the Energy Method when There Is Friction

## VI. 5.6: SIMPLE MACHINES AND MECHANICAL ADVANTAGE

### A. Simple machines allow us to accomplish things with less force; best where there are only conservative forces and no energy is lost to friction, heat

### B. Mechanical Advantage = (effort distance)/(resistance distance)

1. *MA = d _{effort}/d_{resistance}*

2. This quantifies precisely how much less force is required when using simple machines

### C. EFFICIENCY

Measures the degree to which friction and other facts reduce work output from theoretical maximum

1. *Efficiency (%) = W _{output}/Energy_{input}*

## VII. 5.7: THERMAL ENERGY AND MODES OF HEAT TRANSFER

### A. Heat is thermal energy and can be transferred in 3 ways:

1. **CONDUCTION** – direct transfer of heat from one object to another caused by temperature difference

2. **CONVECTION** – transfer of energy due to large scale motion of fluid (or air)

3. ** RADIATION** – absorption of energy from EM radiation

## VIII. 5.8: MOMENTUM

Product of mass and velocity (kg⋅m/s), is a vector

### A. Momentum (p) = mv

### B. IMPULSE (J) – can be used to predict the effect of a force

1. F = ma → a = Δv/Δt → F = mΔv/Δt → FΔt = mΔv = Δmv = Δp

2. FΔt = Δp

### C. Impulse Momentum Theorem

1. J = impulse = change in momentum (remember signs! if a baseball changes direction and speed, one of the speeds must be negative)

2. **J = FΔt =mΔv = Δp**

3. **Δp = p _{f} – p_{initial}** (only use this if collision is elastic)

4. If there is a graph of F (vertical axis) and ∆T (horizontal axis), the area under the curve is the impuse

### D. Conservation of Momentum

1. Consider action/reaction pair

a) F_{1-on-2} is force object 1 exerts on object 2

(1) Impulse = J_{1-on-2} = F_{1-on-2}Δt

b) F_{2-on-1} is force object 2 exerts on object 1

(1) Impulse = J_{2-on-1} = F_{2-on-1}Δt

c) Since the forces are equal and opposite, so are the impulses

d) if J_{1-on-2} = -J_{2-on-1} → Δp_{1} = – ∆p_{2} → ∆p_{1} + ∆p_{2} = 0

e) Law of conservation of momentum: ∆p_{system} = 0 → total p_{initial} = total p_{final}

### E. Collisions

1. Conservation of momentum applies to collisions; total kinetic energy, not necessarily!

2. Consider cue ball hitting 8 ball; initial momentum is cue ball, final momentum is 8 ball plus cue ball after collision

3. Collision definitions:

a) Elastic collision – total momentum and total KE are conserved

b) Inelastic collision – total momentum is conserved but KE is not conserved

c) Perfectly inelastic – an inelastic collision in which the objects stick together afterwards

4. Remember, KE = ½mv^{2}

## IX. 5.9: A NOTE ON ANGULAR MOMENTUM

### A. Moment of inertia is the rotational analog of mass

### B. Torque is the rotational version of force

### C. Angular momentum is the rotational version of linear momentum (think of a rotating frisbee)

1. Angular momentum must be defined relative to some reference point (often axis of rotation)

### D. L = ℓmv = Iω

1. L = angular momentum

2. ℓ = lever arm (shortest distance perpendicular to rotating mass)

3. I = moment of inertia

4. ω = angular velocity (radians/sec) → analog to speed, v

### E. τ = ∆L/∆t

1. Torque is the rate of angular momentum

**CHAPTER 5 SUMMARY**

- Work →
**W = Fdcosθ** - Power
**→ P = W/t** - Kinetic Energy →
**KE = ½ mv**^{2} **W**_{tot}= ∆KE- Gravitational potential energy →
**∆PE**_{grav}= -W_{byF grav}= mg∆h- Gravity is a conservative force
- Friction is NOT a conservative force

- Total Mechanical Energy
*→*E = KE + PE - Conservation of Total Mechanical Energy →
**KE**_{i}+ PE_{i}= KE_{f}+ PE_{f} - Simple Machines:
- Mechanical Advantage =
**d**_{effort}/d_{resistance} - Efficiency (%) =
**W**_{output}/Energy_{input}

- Mechanical Advantage =
- Momentum →
**p = mv**- Impulse-momentum theorem →
**J = ∆p = F∆t** - Conservation of Total Momentum →
**total p**_{initial }= p_{final } - Collisions always conserve momentum
- Elastic collisions also conserve KE
- Inelastic collisions do NOT conserve KE
- Perfectly inelastic collisions lose most of KE (objects stick together)
- Angular Momentum →
**L = ℓmv = Iω**

- Impulse-momentum theorem →