As you move forward, you will notice that many of the problems deal with work, mechanical advantage, and efficiency all at that same time. See if you can answer the following question in its multiple parts.
A pulley system is used to lift a 2000 N grand piano 3 meters. The mover applies 300 N of force in the process.
Mechanical Advantage
Work Output
Work Input
Efficiency
Calculate the mechanical advantage of the pulley system.
\(\mathsf{ F_{input} = 300 \text{ N} }\)
\(\mathsf{ F_{output} = 2000 \text{ N} }\)
\(\mathsf{ M.A. = ? }\)
\(\mathsf{ M.A. = \frac{F_{output}}{F_{input}} }\)
\(\mathsf{ M.A. = \frac{2000 \text{ N}}{300 \text{ N}} }\)
\(\mathsf{ M.A. \approx 7 }\)
How much work does the machine do?
\(\mathsf{ F_{machine} = 2000 \text{ N} }\)
\(\mathsf{ d = 3 \text{ m} }\)
\(\mathsf{ \theta = 90° }\) (implied)
\(\mathsf{ W_{machine} = ? }\)
\(\mathsf{ W = Fd \sin \theta }\)
\(\mathsf{ W = (2000 \text{ N})(3 \text{ m})(\sin 90°) }\)
\(\mathsf{ W = 6000 \text{ Nm} }\)
If the mover pulls 27 meters of rope to lift the piano, how much work does he do?
\(\mathsf{ F_{mover} = 300 \text{ N} }\)
\(\mathsf{ d = 27 \text{ m} }\)
\(\mathsf{ \theta = 90° }\) (implied)
\(\mathsf{ W_{mover} = ? }\)
\(\mathsf{ W = Fd \sin \theta }\)
\(\mathsf{ W = (300 \text{ N})(27 \text{ m})(\sin 90°) }\)
\(\mathsf{ W \approx 8000 \text{ Nm} }\)
What is the efficiency of the pulley system?
\(\mathsf{ W_{out} = 6000 \text{ Nm} }\)
\(\mathsf{ W_{in} = 8000 \text{ N} }\)
\(\mathsf{ eff. = ? }\)
\(\mathsf{ eff. = \frac{W_{out}}{W_{in}} }\)
\(\mathsf{ eff. = \frac{6000 \text{ Nm}}{8000 \text{ N}} }\)
\(\mathsf{ eff. = 75 \text{%} }\)
