1. Power

2. Uphill
You (m = 60 kg) hike up a 30° hill with a net height increase (h = 50 m). What work is done by gravity?
Distance d = h / sinq = (50 m) / sin 30° = 100 m
Work done by gravity W = -mg d sinq =
-(60 kg)(9.8 m/s2)(50 m) = -30 kJ
d = 100 m
Gravity does negative work on the hiker
-mg

3. Rate of Work
You (60 kg) walk up a 30° hill with a net height increase of 50 m at 1 m/s.
t = (50 m)/sin30°/(1 m/s) =100 s
W = 30 kJ
You run up the same 30° hill with a net height increase of 50 m at 4 m/s.
t = 25 s, W = 30 kJ
But, running is harder!
The rate of work has increased by 4 times by running.
Hiker does positive work to overcome gravity
d = 100 m

4. The Watt The rate of work is called power.
The SI unit of power is the watt (W).
1 watt = 1 J/s = 1 N m/s = 1 kg m2 / s3
Energy can be measured in watt-seconds = joules.

5. Average Power
The walker had an average power output based on the work compared to the time.
P = W / t
P = 30 kJ / 100 s = 300 W
The runner generated the same work in one quarter of the time.
P = 30 kJ / 25 s = 1200 W
When running seems harder, it isn’t work, it’s power.

6. Instantaneous Power Work does not have to be uniform over time.
Moving over a series of hills and valleys (changing work)
Walking and running (changing rate)
The power expended at one instant is the limit of work done over a very small time interval.

7. Power Plants Electrical power is measured in watts.
60 W light bulb
1000 MW power plant
Energy used is measured in power times time.
If electricity costs $ per kWh, how much does it cost to leave a 1500 W floodlamp on all year?
Energy used is W = Pt = (1500 W)(3.2 x 107 s) = x 1010 J
Cost is (4.8 x 1010 W s) * (1 kW / 1000 W) * (1 h / 3600 s)*(0.083 / kWh) = $1,100

8. Force and Velocity Power is work per time.
Work is force acting over a distance
Distance per time is velocity
Power is force times velocity. v F

9. Hiking Speed
You (60 kg) walk up a 30° hill with a net height increase of 50 m at 1 m/s.
F =(60 kg)(9.8 m/s2)= 600 N
v = 1 m/s
q = 120°
P =(600 N)(1 m/s)(cos 120°)
P = -300 W
v = 1 m/s
Hiker uses power to overcome gravity
mg = 600 N