Power and Efficiency Power is defined as the amount of work performed per unit of time. If a machine or engine performs a certain amount of work, dU, within a given time interval, dt, the power generated can be calculated as P  dU / dt Since the work can be expressed as dU = F • dr, the power can be written P  dU / dt  ( F .dr ) / dt  F .(dr / dt )  F .v Thus, power is a scalar defined as the product of the force and velocity components acting in the same direction. Power Using scalar notation, power can be written where ϴ is the angle between the force and velocity vectors. P  F .v  Fv cos  So if the velocity of a body acted on by a force F is known, the power can be determined by calculating the dot product or by multiplying force and velocity components. The unit of power in the SI system is the Watt (W) where 1W  1 J / s  1( N . m) / s In the FPS system, power is usually expressed in units of horsepower (hp) where 1hp  550( ft lb)/s  746W Efficiency The mechanical efficiency of a machine is the ratio of the useful power produced (output power) to the power supplied to the machine (input power) or e   power output  / ( power input ) If energy input and removal occur at the same time, efficiency may also be expressed in terms of the ratio of output energy to input energy or e   energy output  / (energy input ) Machines will always have frictional forces. Since frictional forces dissipate energy, additional power will be required to overcome these forces. Consequently, the efficiency of a machine is always less than 1 Procedure for Analysis Find the resultant external force acting on the body causing its motion. It may be necessary to draw a free-body diagram. Determine the velocity of the point on the body at which the force is applied. Energy methods or the equation of motion and appropriate kinematic relations, may be necessary. Multiply the force magnitude by the component of velocity acting in the direction of F to determine the power supplied to the body ( P  Fv cos  ) In some cases, power may be found by calculating the work done per unit of time ( P  dU / dt ) If the mechanical efficiency of a machine is known, either the power input or output can be determined. Example (1 of 3) Given: A 50 kilogram block (A) is hoisted by the pulley system and motor M. The motor has an efficiency of 0.8. At this instant, point P on the cable has a velocity of 12 m / s which is increasing at a rate of the pulleys and cable. 6 m / s2. Neglect the mass of Find: The power supplied to the motor at this instant. Plan: 1. Relate the cable and block velocities by defining position coordinates. Draw a FBD of the block. 2. Use the equation of motion to determine the cable tension. 3. Calculate the power supplied by the motor and then to the motor. Example (2 of 3) Solution: 1) Define position coordinates to relate velocities. Here s p is defined to a point on the cable. sA Also is defined only to the lower pulley, since the block moves with the pulley. From kinematics, S P  2s A  1  a P  2a A  0  a A   a p / 2  3m / s 2  3m / s 2 () Draw the FBD and kinetic diagram of the block: Example (3 of 3) 2) The tension of the cable can be obtained by applying the equation of motion to the block.  F v  mAa A 2T  50(9.81)  50(3)  T  320.3 N 3) The power supplied by the motor is the product of the force applied to the cable and the velocity of the cable. Po  F .v  (320.3)(12)  3844 W The power supplied to the motor is determined using the motor’s efficiency and the basic efficiency equation. Pi  Po /   3844 / 0.8  4804 W  4.8 kW