5. Work, Energy and Power

Work, Energy and Power Work done by a force on a body is defined as the product of the average force acting on an object and the displacement moved in the direction of the force. The general case: The force (F) and displacement (s) may not point in the same direction. Only the component of the force along the displacement does work. The work done by F above is given by W = (Fcos ) s Work done by a variable force The work done by F can be obtained by finding the area under the F-s graph. W= ∫ F ds Or estimate the area under, e.g. by counting squares and multiplying by the area of each square. W = area A for a F-s graph W = area B for a s-F graph The work done by a force can be either positive or negative It is positive if the force acts in the same direction as the displacement. It is negative if the force is in the opposite direction to the displacement. When a force does positive work on a body, it increases the energy of the body. When a force does negative work on a body, it removes energy from the body.

Different forms of Energy (1) Mechanical energy: energy that is possessed by an object due to its motion or due to its position. Mechanical energy = Kinetic energy + Gravitational potential energy + Elastic potential energy (2) Energy in a matter: chemical energy, nuclear energy and internal energy. (3) Electromagnetic energy: electric potential energy, magnetic potential energy, and radiant energy Principle of Conservation of Energy Energy cannot be created nor destroyed but it may be transformed (converted) from one form to another.

In general, W = ∫ F dx If F is a conservative force, this work does not change the total energy of the system. This work transforms the energy of the system between kinetic energy and potential energy only. Thus any kinetic energy K gained by such a system must equal to its los s in potential energy U and vice versa. That is, W = dK = –dU [dK: change in K; dU: change in U] ∫ F dx = –dU dx dUF   F = - dU dx is interpreted as: o The magnitude of the F at a point at displacement x is given by the gradient of the U-x graph at that point. o The negative sign mean that F acts in the direction in which potential energy would decrease when an object is moved by the force F.  When applying the equation F = - dU dx , U refers to the type of potential energy that is of the same nature as the force F. That is, F U Elastic force Elastic Potential Energy Gravitational force Gravitational Potential Energy Electric force Electric Potential Energy

Power is the rate of doing work with respect to time (or work done per unit time) or the rate of transfer of energy with respect to time. Average Power Average power = Total work done time taken for this amount of work to be done <P> = W t Instantaneous Power The rate at which work is done might not be constant. On a graph of work done against time, instantaneous power ( dW dt ) is the gradient of the graph at a particular instant, as shown below. Mathematically, the instantaneous power is expressed as: P = dW dt = d(F.s) dt If the force F remains constant with time, P = F d(s) dt = Fv 𝑬𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒄𝒚 = 𝒖𝒔𝒆𝒇𝒖𝒍 𝒆𝒏𝒆𝒓𝒈𝒚 𝒐𝒖𝒕𝒑𝒖𝒕 𝒕𝒐𝒕𝒂𝒍 𝒆𝒏𝒆𝒓𝒈𝒚 𝒊𝒏𝒑𝒖𝒕 𝒙𝟏𝟎𝟎% or 𝑬𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒄𝒚 = 𝒖𝒔𝒆𝒇𝒖𝒍 𝒑𝒐𝒘𝒆𝒓 𝒐𝒖𝒕𝒑𝒖𝒕 𝒕𝒐𝒕𝒂𝒍 𝒑𝒐𝒘𝒆𝒓 𝒊𝒏𝒑𝒖𝒕 𝒙𝟏𝟎𝟎% Time, t Work done, W power = = gradient of W-t graph at a point