
The pulley and the wheel and axle are two other categories of simple machines related to the lever. Pulleys can be used to redirect a force, and combinations of pulleys used in blockandtackle systems can significantly multiply force. The wheel and axle multiplies force by the ratio of the wheel to axle radius. When the axle drives the wheel, the output force is reduced as a tradeoff for increased speed. Losses such as friction cause real machines to have less than 100% efficiency and lower output forces than expected from their ideal mechanical advantage.

tension, pulley, block and tackle, efficiency, ideal mechanical advantage, wheel and axle, gear, gear ratio


$$\eta =\frac{{W}_{o}}{{W}_{i}}$$
 
$$M{A}_{ideal}=\frac{{d}_{i}}{{d}_{o}}$$
 
$$M{A}_{wa}=\frac{{r}_{w}}{{r}_{a}}$$
 
$$M{A}_{g}=\frac{{\tau}_{o}}{{\tau}_{i}}=\frac{\text{outputteeth}}{\text{inputteeth}}$$
 

Review problems and questions 

 Describe the measurements and calculations that you need to make to determine the mechanical advantage of a
 lever,
 block and tackle,
 wheel and axle.

 For the lever, you need to measure (i) the length of the input arm (between the fulcrum and the location of the input force) and (ii) the length of the output arm (between the fulcrum and the location of the output force). The mechanical advantage of a lever is calculated by dividing the length of the input arm by the length of the output arm.
 For the block and tackle, you need to count the number of ropes supporting the output load. The mechanical advantage is equal to that number.
 For the wheel and axle, you need to measure (i) the radius of the wheel and (ii) the radius of the axle. If the input force is applied to the wheel, then the mechanical advantage is calculated as the ratio of the wheel radius to the axle radius.

 A force is applied to a large gear with 51 teeth. That large gear then turns a smaller gear with only 17 teeth.
 What is the gear ratio of the two gears?
 What is the mechanical advantage of the two gears?
 If the small gear instead were to turn the large gear, what is the gear ratio?
 Likewise, what is the mechanical advantage when the small gear turns the large gear?

Answer:  3.0
 0.33
 0.33
 3.0
Solution: The gear ratio is the ratio of input teeth to output teeth:$$GR=\frac{\text{inputteeth}}{\text{outputteeth}}=\frac{51}{17}=3.0$$
 The mechanical advantage is the ratio of output teeth to input teeth:$$MA=\frac{\text{outputteeth}}{\text{inputteeth}}=\frac{17}{51}=0.33$$
 When the gears act in the opposite direction, the output gear becomes the input gear and vice versa. The gear ratio is now the inverse of the previous value, or 0.33.
 The mechanical advantage is now the inverse of the previous value, or 3.0.

 In a car, when the steering wheel is turned it causes the steering column (the axle) to turn. Why wouldn’t you want to design the car to simply have a steering axle? Use forces in your answer.

In a wheel and axle machine, smaller input forces are required to turn the wheel than are required to turn the axle. If you had a steering axle, then larger forces would be required to turn it—and it would be very difficult to steer!

 In a winch, a crank with a radius of 50 cm is turned to cause a rope to wrap around a drum with a radius of 10 cm.
 What is the ideal mechanical advantage of this machine?
 If the winch had friction that caused the efficiency to be 60%, what would be its mechanical advantage?

Answer:  5
 3
Solution:  The ideal mechanical advantage comes from the ratios of the wheel radius (the crank) to the axle radius (the drum):$$M{A}_{ideal}=\frac{{r}_{w}}{{r}_{a}}=\frac{50\text{cm}}{10\text{cm}}=5$$
 The (actual) mechanical advantage of the system is proportional to the output work, which is only 60% of the ideal amount of output work. So the mechanical advantage is 60% of the ideal case, or 0.6×5 = 3.

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