Strand B. Energy Unit 3. Power Contents Page Power 2 Power in the Home 5
B.3.1 Power In this unit we have considered energy and energy transfer. We have also learnt that energy transferred to an object is equal to the amount of work done on that object. For example, if we were to lift a 10kg barbell through a vertical distance of 1.5m, we would do work. The energy transferred to increase the barbell s GPE would be mgh = 10kg 10ms -2 1.5m = 150J, and as such we would have performed 150J of work on the barbell. However, this definition of energy transfer, or work done, makes no reference to time. If we took 1 second, 1 hour, or even 1 week to lift the barbell, we would have transferred the same amount of energy, or accomplished the same amount of work. Sometimes it is important to know how fast energy is transferred by doing work. For example, we know that there is a difference between completing a 5km run at a comfortable pace, and trying to cover that same distance at the speed of a 100m sprinter (even though to first approximation, the work done would be the same). We describe the rate of energy transfer, or the rate of doing work in terms of power. Power is the rate of energy transfer or the rate of doing work, and is a measure of energy transferred per second. PPPPPPPPPP PP = eeeeeeeeeeee tttttttttttttttttttttt tttttttt tttttttttt = wwwwwwww dddddddd tttttttt ttaaaaaaaa or in symbols PP = EE tt = WW tt Since energy and work are both measured in Joules, and time is measured in seconds, the SI unit of power is the Joule per second (Js -1 ). We call 1Js -1 the Watt, (W) after the English inventor James Watt. The non-si unit of horsepower often used as a measure of the power of cars comes from James Watt s experiments. He measured that a horse could do 33,000 foot-pounds of work per minute (that s 746 W) lifting coal from a coal mine. You may have seen the Watt before, since many household appliances are power rated in Watts. For example, a light bulb rated at 100W transfers 100J of electrical energy per second into light and heat. A lift with a power rating of 6000W transfers 6000J of gravitational potential energy to the lift every second. 2
Power is, like energy, a scalar quantity. Some typical power outputs can be very large and are listed in Table 3.1.1. Due to the magnitude of power output for some processes, power is often given in kw (1kW (kilowatt) = 1,000W), MW (1MW (megawatt) = 1,000,000W) and GW (1GW (gigawatt) = 1,000,000,000W). As such, one should get used to using prefixes, and standard form to avoid simple calculation error. Table 3.1.1. Typical Device / Process Torch Electric light bulb Microwave oven High speed train Nuclear power station Space shuttle on launch Our Sun Typical Power Output 1W 100W 1000W = 1kW 1,600,000W = 1.6MW 1,000,000,000W = 1GW 27500000000W = 27.5GW 100,000,000,000,000,000,000W = 100,000PW (peta) Worked Example Dave is an amateur triathlete and has been asked to take part in a charity fund raising event. It entails running up the stairs of the Shard building (the tallest building in the UK at 309m). Dave, who weighs 72kg, has set a personal goal of 12 minutes to complete the run. What must his average power output be in order to achieve his goal? Answer: Dave wants to complete the run in t = 12min = 12 60=720s, lifting his mass m = 72kg to a height of h = 309m. Since in this instance, the work Dave has to do is against the gravitational field of the Earth, W = mgh and; PP = EE tt = WW tt = mmmmh tt = 72kkkk 10mmss 2 309mm 720ss = 309WW In the above example, the energy transferred (or the work done) is against gravity. Often however, mechanical work involves a force F that moves an object a displacement s, and the work done must be calculated using W = FΔs (more on 3
this in Strand C). Since power is the work done over a time t, and distance s / time t is equal to velocity; PPPPPPPPPP = WWWWWWWW DDDDDDDD TTTTTTTT = FFFFFFFFFF DDDDDDDDDDDDDDDD TTTTTTTT = FFFFFFFFFF VVVVVVVVVVVVVVVV PP = ΔWW Δtt or in symbols = FF Δss Δtt = FFFF Worked Example An aeroplane engine generates a driving force (thrust) of 120000N. When flying at 700km/h, what power does the engine develop? Answer The engine develops F = 120000N in the direction of the planes velocity v = 700km/h = 700000m/3600s = 194m/s. Thus PP = FFFF = 120000NN 194mmss 1 = 23,280,000WW = 23.28MMMM Exercise B.3.1. 1. At the time of writing, the recommended calorific intake by the NHS for a woman to maintain bodyweight is 8400kJ (2000 calories) per day. According to the NHS, what is the average power output of the female body? 2. During the snatch event, an Olympic power lifter can lift 200kg through 2.2m in a time of 4 seconds. How much power does the power lifter generate? 3. A crane develops 18kW of lifting power. What is the shortest possible time that the crane could lift a 5000kg concrete block to a height of 12m? 4. A car engine develops 200kW of power when travelling at a constant 35m/s. What force does the car s engine generate at the tyres? 5. In September 2015, asteroid 2015SK7 passed within 26,600km of the Earth (a very small distance in astronomical terms (for example the Earth Moon distance is 384,400km). 2015SK7 has a diameter of only 6m, but if it collided with the Earth, it would have had a mass on impact of 270,000kg and an impact velocity of 16.5km/s. If the impact time was 6s, what would be the power 4
generated at impact? Compare this to the global energy usage of 4.5 10 16 J every second. Challenge Question 6. A pump is required to lift 1000kg of water per minute from a 12m deep well. If the pump ejects the water at 20m/s, (a) how much work is done lifting the water? (b) how much work is done providing the KE to the water? (c) what is the minimum power rating for the pump? B.3.2. Power in the Home Whenever an appliance such as a kettle or a washing machine is used, energy is transferred from the mains electricity supply to the device, enabling it to do work. The energy supplied per second may be different for each device, but all electrical appliances are labelled with a power rating in a similar way to that shown by Figure 3.2.1. A power rating of 2000W signifies that 2000J of Figure 3.2.1 electrical energy is transferred to the device every second. For any electrical device, the total energy E supplied depends on; The time t that the device is on The power rating P of the device (the energy supplied per second) To calculate the total energy used by an electrical device EEEEEEEEEEEE uuuuuuuu = PPPPPPPPPP RRRRRRRRRRRR tttttttt or in symbols EE = PP tt 5
Worked Example A vacuum cleaner is rated at 2200W. It takes 7 minutes to vacuum the living room. How much electrical energy was used vacuuming the living room? Answer t = 7 minutes = 7 60s = 420s P = 2200W The total electrical energy used is the power of the device (energy transferred per second) multiplied by the time the device is on; EE = PPPP = 2200WW 420ss = 924,000JJ In the above worked example, the vacuum cleaner used 924,000J of electrical energy (or 924kJ) in only 7 minutes. In general, domestic power usage if measured in Joules is a large number, because one Joule is a relatively small amount of electrical energy. As such, electricity and other power companies sell energy in a non-si unit of kilowatt-hours (kwh). The kwh is not a derived SI unit since the SI unit of time is the second, not the hour, but it is an accepted unit of energy. It is a common student misconception that the kwh is a unit of power, when actually the kwh is a unit of energy. 1kWh = 1000Js -1 1 hour Energy = Power time For example A 1kW electric heater that is switched on for 1 hour uses 1kW 1hour = 1kWh of electrical energy. A 1kW electric heater switched on for 10 hours uses 1kW 10 hours = 10kWh of electrical energy. 6
Worked example Alison is cooking Christmas dinner for her family. She puts the turkey in the oven and calculates from the weight of the bird that it requires 5 hours of cooking at an average temperature of 220 C. At this temperature, the oven rating is 1900W. How much electrical energy in kwh, and then in Joules, does it take to roast the turkey? Answer The turkey takes t = 5 hours at P = 1900W = 1.9kW EE = PPPP = 1.9kkkk 5hoooooooo = 9.5kkkkh In Joules, we must use time in seconds, and power in watts. Therefore t = 5 hours = 5 60 60=18000s and P = 1900W EE = PPPP = 1900WW 18000ss = 34,200,000JJ = 34.2MMMM note here that 1kWh = 1000Js -1 60 60s = 3600000J = 3.6MJ Electricity meters in the home as shown by Figure 3.2.2 record the electrical energy used within the whole house, irrespective of the appliance used, providing a reading in kwh to the power company. The power company can then calculate how much to charge the household. Commonly, the electricity bill is calculated over a 3-month period, and the difference in meter reading at the start and end of this time period shows the energy supplied in kwh. The electricity bill is calculated using; total cost = number of kwh used cost per kwh Figure 3.2.2 If the standard rate from a supplier for electricity is 15.3pence per kwh, and an average household uses 825kWh of electrical energy every 3 months (per quarter), one can expect a quarterly electricity bill of; 825kWh 15.3p = 12622.5p = 126.23 7
Exercise B.3.2 1. An electric light bulb used for 5 hours a day is changed from a standard 100W bulb to an energy efficient 12W bulb. How many Joules of energy is saved per day? 2. A microwave is used to heat a ready meal. The instructions state; Pierce lid. Microwave on full power for 3 ½ minutes. Remove lid and stir, then heat for a further 2 minutes. If the microwave uses 495kJ of electrical energy to heat the ready meal, what is the power rating of the microwave? 3. If left on standby, the average LCD television consumes 22kWh of electrical energy per year. How many Joules of energy are there in 22kwh? 4. The meter man visits Jim and Jayne to read the electricity meter. He informs Jayne that the reading on the 1 st of January was 29500kWh and on the 1 st of April was 30400kWh. If the current price of electricity is 15.5 pence per kwh, what quarterly bill can Jim and Jayne expect? 5. A combination fridge-freezer uses on average 250W and must be left on 24 hours per day. If the cost of electricity is 15.5 pence per kwh, what is the annual running cost of the fridge-freezer? Challenge Question 6. The table below shows average power consumption and annual running time for the appliances in a Kitchen. Appliance Power Capacity Annual Running Time Fridge-freezer 250W Continuously Dish washer 1200W 1 hour, 5 times per week Washing machine 2.5kW 3 times per week for 120 minutes Electric oven 2200W 6 ½ hours per week Calculate the annual cost in electricity at 15.3p per kwh. How many Joules of energy are used in the kitchen per day (on average)? 8