Gravitational Potential Energy


 Ernest Hawkins
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1 Gravitational Potential Energy Consider a ball falling from a height of y 0 =h to the floor at height y=0. A net force of gravity has been acting on the ball as it drops. So the total work done on the ball is: W tot = F y y = (mg) (yy 0 ) = (mg)(h) =mgh but W tot = E k = ½ mv 2 0 so mgh = ½ mv 2 and the work done by gravity equals the kinetic energy gained by the ball. Call the gravitational potential energy E p and define it to be: E p = mgy Realize that only changes in potential energy can be measured. Above is valid so long as g remains constant over the range of distance represented by y. (Really y with y the distance dropped)
2 From your text page 615: A few comments should be made here about the selection of the y=0 position. Gravitational potential energy is a relative quantity, always measured with respect to some arbitrary position where y=0. In most cases, it does not matter where y=0 is chosen, but it is usually easiest to choose y=0 at the lowest possible position in the diagram (for example at floor level). This ensures that all yvalues and all gravitational potential energies will be positive (or zero) and will be easy to work with. We shall see that what is usually important is the change in the gravitational potential energy, which does not depend on where we have chosen y=0. In the defining equation for E p, mg is positive (because it is the magnitude of a vector not the vector itself) and the potential energy increases with elevation. Remember, energy is not a force and is not a vector.
3 Conservation of Energy Energy can be converted into different forms but it can be neither created nor destroyed. When you burn a log to create heat, you are changing chemical energy into heat energy. The only seeming exception to the above law relates to nuclear energy whereby some mass is converted into energy. However, at a fundamental level, mass and energy are two manifestations of the same thing (it has no name!) and this thing is conserved. It is convenient and correct to call this thing energy. Note that conservation of energy differs from energy conservation, which refers to the careful use of energy resources. Conservation of energy is a law of nature; energy conservation is something that we should all practise. Energy conservation is a bit of a misnomer as nature does that for us. What we are conserving is energy in a particularly useful and concentrated form.
4 The law of conservation of energy can be used to solve problems that kinematics alone could not solve. Kinetic plus potential energies at the start of the problem must be the same at the end of the problem. Or Total energy (start) = Total energy (end) KE 1 + PE 1 = KE 2 + PE 2 ½ mv mgy 1 = ½ mv mgy 2 Consider sample problem 65. The units for energy can be found from either the kinetic or potential energy formulas. 2 m m kg = kg m 2 2 s s = Newton m Joule
5 Work done by Friction The work done by friction is the kinetic friction force multiplied by the distance moved. This work always produces heat or thermal energy. W f = F k r E th = F k r The friction force is always opposite to the velocity and will also be opposite to the displacement. Hence the work is done by the body not on the body and the work is negative. The body loses energy because of friction. The lost energy goes into heating the surroundings.
6 Efficiency of Energy Conversions efficiency = useful energy out total energy in Power Power is the rate of doing work. It is also the rate at which energy is being consumed. It is simply P = E/t The unit for power is watt which is a joule per second. Kilowatt, or kw, is a common power unit for electricity. Because your house is metred to measure kilowatts, your rate of energy consumption, the amount of energy you use is given by the kilowatts multiplied by the time. Rather than use joules which is a rather small unit of energy, the power company uses kilowatthours. There are 3600 seconds in an hour and 1000 watts in a kilowatt so there are 3.6 million joules in a kilowatthour.
7 Momentum Chapter 7 Momentum is defined as the product of the object s mass and velocity. It is a vector quantity, one that is useful for problem solving. The symbol for momentum is p. p = mv Of course this means that p x = mv x and p y = mv y Even as work is defined as the force times a distance, so impulse is defined as force times time. An impulse applied to an object changes its momentum. We write: F t = p Consider motion in the xdirection: F x t = mv 2x mv 1x ma x t = mv 2x mv 1x or a x t = v 2x v 1x
8 You recognize this as v x = v 0x + a x t Note that the dimensions of Force time is Newtonsecond. The dimensions of momentum are mass velocity or kg m/s. But a Newtonsecond is a: m m kg s kg s 2 = s and we get the same either way as we must. Forces involved in collisions are usually very large. Consider a baseball bat striking a ball, as in sample problem 71.
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