Essential Physics

Section 2 review

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 block-and-tackle 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. Read the text aloud

Read the text aloud

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

η = \frac{W_{o}}{W_{i}}

M A_{i d e a l} = \frac{d_{i}}{d_{o}}

M A_{w a} = \frac{r_{w}}{r_{a}}

M A_{g} = \frac{τ_{o}}{τ_{i}} = \frac{output teeth}{input teeth}

Review problems and questions

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

A force is applied to a large gear with 51 teeth. That large gear then turns a smaller gear with only 17 teeth.
1. What is the gear ratio of the two gears?
2. What is the mechanical advantage of the two gears?
3. If the small gear instead were to turn the large gear, what is the gear ratio?
4. 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: $G R = \frac{input teeth}{output teeth} = \frac{51}{17} = 3.0$
The mechanical advantage is the ratio of output teeth to input teeth: $M A = \frac{output teeth}{input teeth} = \frac{17}{51} = 0.33$
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 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.
1. What is the ideal mechanical advantage of this machine?
2. 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): $M A_{i d e a l} = \frac{r_{w}}{r_{a}} = \frac{50 cm}{10 cm} = 5$
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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