NJC 2023 Work, Energy, Power Notes

National Junior College 2023 Work, Energy and Power - 1 - Work, Energy and Power Contents ▪ Work ▪ Kinetic Energy & Potential Energy ▪ Energy conversion and conservation ▪ Power and Efficiency Learning Outcomes Candidates should be able to: a) define and use work done by a force as the product of the force and displacement in the direction of the force b) calculate the work done in a number of situations including the work done by a gas which is expanding against a constant external pressure: W p V= c) give examples of energy in different forms, its conversion and conservation, and apply the principle of energy conservation. d) show an appreciation for the implications of energy losses in practical devices and use the concept of efficiency to solve problems. e) derive, from the equations for uniformly accelerated motion in a straight line, the equation 212kE mv= . f) recall and use the equation 212kE mv= . g) distinguish between gravitational potential energy, electric potential energy and elastic potential energy. h) deduce that the elastic potential energy in a deformed material is related to the area under the force-extension graph. i) show an understanding of an d use the relationship between force and potential energy in a uniform field to solve problems. j) derive, from the definition of work done by a force, the equation pE mgh= for gravitational potential energy changes near the Earth’s surface. k) recall and use the equation pE mgh= for potential energy changes near the Earth’s surface. l) define power as work done per unit time and derive power as the product of a force and velocity in the direction of the force.

National Junior College 2023 Work, Energy and Power - 2 - 1 Work 1.1 Work done by a constant force For a constant force F, where F is the magnitude of the constant force x is the displacement  is the angle between the force and displacement You can either view this as resolving F along the displacement ( F cos θ) or resolving x along the force (x cos θ). The S.I. unit for work done is joule (J). Example 1(a) Find the amount of work done by the tension of the cable when the crane pulls a load 20 m horizontally with a tension of 2000 N, making an angle of 30 o with the horizontal. Solution cos (2000)(20) cos30 26000 J W Fx = = = Example 1(b) Find the amount of work done by the tension of the cable when the crane lifts a load 20 m vertically with a tension of 2000 N. Solution cos (2000)(20) cos 0 40000 J W Fx = = = Definition Work done by a force on the body is defined as the product of the force and the displacement in the direction of the force. cosW Fx = F x θ Load v F a) define and use work done by a force as the product of the force and displacement in the direction of the force

National Junior College 2023 Work, Energy and Power - 3 - Example 1(c) Find the amount of work done by the tension of the cable when the crane lowers a load vertically down 20 m with a tension of 2000 N. Solution cos (2000)(20) cos180 40000 J W Fx = = =− Discussion Work can be positive or negative. • Positive work is done by a force when a non-zero component of the force exerted is in the same direction as the displacement of the object. • Negative work is done by a force when a non-zero component of the force exerted is in the opposite direction to the displacement of the object. Example 2 A communications satellite moves in a circular orbit at a constant speed in response to gravity. Which of the following statements is correct? A The earth does positive work on the satellite. B The earth does negative work on the satellite. C The earth does no work on the satellite. D Once a coordinate system is specified, the work changes sign every half orbit, so that the average work is zero. Other cases where work done is zero: • Spacecraft travelling at constant velocity Work done by the net force on spacecraft = 0, because net force = 0 • Force exerted on an immovable object Work done by the force applied on the wall = 0, because displacement = 0 Earth Satellite Load F v

National Junior College 2023 Work, Energy and Power - 4 - 1.1.1 Work done by a gas expanding against constant pressure There is another special example of work done by a constant force: work done by a gas expanding against constant pressure (to be covered in more detail in Thermal Physics). Consider the expansion of a gas in a cylinder with a massless frictionless piston separating it from the surrounding. If the surrounding is a large entity (e.g. atmosphere), during the expansion, the external pressure Pext, and hence the external force Fext by the surrounding on the piston, is effectively constant. F is the force by the gas on the surrounding. Therefore, during this process when the piston moves over a distance ∆x, Work done by gas on surrounding (workdone by F) = F∆x Work done by F is positive as the displacement and force is in the same direction Work done by surrounding on gas (Work done by ) ext ext ext ext F F x P A x PV =−  =−  =−  The work done is negative because the external force and the displacement of the piston are in opposite directions. The expression above gives us the work done by the surrounding on the gas. The negative sign shows that energy is transferred from the gas to the surrounding. The more common term to describe this process is work done by the gas on th e surrounding, which is simply the negative of work done by the surrounding on the gas. A: cross-sectional area of the piston ∆V: change of volume of the gas ∆V ∆x Pext Fext A b) calculate the work done in a number of situations including the work done by a gas which is expanding against a constant external pressure: W p V=

National Junior College 2023 Work, Energy and Power - 5 - 1.2 Work done by a force of varying magnitude We have been considering situations where the force acting is of a constant magnitude in the direction of displacement. In many practical situations, the force acting on an object varies. Suppose an object is being displaced along the x-axis under the action of a force Fx that acts in the x direction and varies with position. Let’s imagine we were able to use a force sensor and a data logger to measure this variation , and obtain the plot shown below. Consider the part when the object is displaced in the direction of increasing x from x = xi to x = xf. The total displacement x can be calculated as the sum of many very small step s x. During each small step, the force is practically constant. The total work done = sum of work done for each step W = W1 + W2 + W3 + …… = F1x + F2x + F3x + ….. Expanding against constant pressure, Work done by a gas on surrounding = negative work done by surrounding on gas = - (-Pext ∆V) = Pext ∆V

National Junior College 2023 Work, Energy and Power - 6 - As the width of the small steps, x, gets really small, the sum of area of these strips will just be the area under the graph of Fx against x. f i x x x W F dx= 2 Energy Energy is measured using the same unit as work done, namely joule (J). Energy and work are two closely linked concepts. Work is a process of conversion of energy from one form to another form or the transfer of energy from one body to another by means of a force. 2.1 Kinetic Energy Ek Derivation of the equation 21 2 kE mv= The starting point is to define that the kinetic energy of a body at velocity v is the work done on it by an external force F to bring it from rest t