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Power. 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 / sin q = (50 m) / sin 30 ° = 100 m Work done by gravity W = - mg d sin q = - (60 kg)(9.8 m/s2)(50 m) = - 30 kJ. d = 100 m.

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Power

Power


Uphill

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


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/ms) =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.

Rate of Work

Hiker does positive work to overcome gravity

d = 100 m


The watt

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.


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.

Average Power


Instantaneous power

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.


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 $ 0.083 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) = 4.8 x 1010 J

Cost is (4.8 x 1010 W s) * (1 kW / 1000 W) * (1 h / 3600 s)*(0.083 / kWh) = $1,100

Power Plants


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.

Force and Velocity

v

F

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