How is power defind?
▪ Power – The rate at which work is done or energy is transferred.
*Units of power – Watts (W)
*Power is sometimes given in horsepower; 1 horsepower = 746 W
▪ Efficiency – \(\varepsilon =\frac{P_{used}}{P_{supplied}}\)
Example A: An elevator lifts twenty 70 kg people 3.0 m in one minute.
a) How much power is required?
Answer/Explanation
Ans: \(P=\frac{W}{t}=\frac{Fd}{t}=\frac{mgd}{t}\)
\(\Rightarrow P=\frac{20(70 Kg)(10 m/s^{2})(3m)}{60s}=700 W\)
b) If the motor powering the elevator is 36% efficient, how much power needs to be supplied?
Answer/Explanation
Ans: \(\varepsilon =36=\frac{700 W}{P_{supplied}}\Rightarrow P_{supplied}=1944 W\)
Example B: Find the instantaneous power to a 5.0 kg mass at time t = 5.0 s given that 𝑥(𝑡) = 5.0sin(0.2𝑡)
Answer/Explanation
Ans: v(t) = x'(t) = cos (.2t)
a(t) = v'(t) = -2sin(.2t)
F(t) = ma(t) = -sin(.2t)
P=F(t). v(t) = – sin(.2x(5s)) • cos(.2x(5s))= -.45 W
Example C: A golf cart’s battery supplies 1200 W of power and is 80% efficient. The cart travels at constant speed at 9.0 m/s. The mass of the cart and rider is 350 kg. What is the force of friction on the cart?
Answer/Explanation
Ans: *The golf cart is treated as an object that is sliding in this question
Since 1200 W is supplied and the motor is 80%, only 960 W is actually used to move the cart.
\(\sum F_{x}=0=F_{MOTOR}-F_{f}\)
\(\rightarrow F_{MOTOR}=F_{f}\)
Since the cart moves at constant speed, the sum of the forces in the horizontal direction is zero.
𝑃𝑢𝑠𝑒𝑑 = 𝐹𝑀𝑂𝑇𝑂𝑅𝑣
Since the speed of the cart is constant, power can be set equal to force times velocity (this is only true for instantaneous power or when velocity is constant). Since the force of the motor and the force of friction are equivalent the following substitution can be made to solve for the force of friction:
