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Power

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

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  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. 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. Rate of Work 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. 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

  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. 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

  8. 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

  9. 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 Hiking Speed v = 1 m/s Hiker uses power to overcome gravity mg = 600 N

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