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# lecture+notes+10 - 14:440:222 Dynamics(Lecture Note#10...

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14:440:222 Dynamics (Lecture Note #10) Instructor: Prof. Peng Song Rutgers University Today’s Lecture Power and efficiency Conservative forces and conservation of energy

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Engines and motors are often rated in terms of their power output. The power output of the motor lifting this elevator is related to the vertical force F acting on the elevator, causing it to move upwards. Given a desired lift velocity for the elevator (with a known maximum load), how can we determine the power requirement of the motor? Power and Efficiency: Applications The speed at which a truck can climb a hill depends in part on the power output of the engine and the angle of inclination of the hill . For a given angle, how can we determine the speed of this truck, knowing the power transmitted by the engine to the wheels? Can we find the speed, if we know the power? If we know the engine power output and speed of the truck, can we determine the maximum angle of climb of this truck ? Power and Efficiency: Applications (cont’d)
Thus, power is a scalar defined as the product of the force and velocity components acting in the same direction. Since the work can be expressed as dU = F d r , the power can be written P = dU/dt = ( F d r )/dt = F (d r /dt) = F v 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 Power is defined as the amount of work performed per unit of time. Power and Efficiency Using scalar notation, power can be written P = F v = F v cos θ where θ is the angle between the force and velocity vectors . In the FPS system, power is usually expressed in units of horsepower (hp) where 1 hp = 550 (ft · lb)/s = 746 W . 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 1 W = 1 J/s = 1 (N · m)/s . Power

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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 ε = (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 .
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