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A compound machine is a machine that is made up of two or more simple machines that work together. All compound machines, from a Roman crane to the modern automobile, can be broken down into different components that fall within one of the six kinds of simple machines. Even the human body can be modeled as a compound machine; each joint in the human arm and leg acts as either a first- or third-class lever, depending on which muscle is being used. The bicycle is a compound machine. Its drivetrain is composed of a front wheel-and-axle system called the crank, a chain connecting the front and back gears, and a rear wheel-and-axle system. 
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compound machine, biomechanics
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Review problems and questions |
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- During the dissection of a human cadaver, a medical student makes the following measurements: The biceps are attached 1.25 cm below the elbow; the triceps are attached 0.8 cm above the elbow; the upper arm is 30 cm long; and the distance between the elbow and middle fingertip is 45 cm.
- What was the ideal mechanical advantage of the forearm when contracting the arm?
- What was the ideal mechanical advantage of the forearm when extending the arm?
- If, during a range of motion extending the arm, the upper arm had a mechanical advantage of 0.15, what was the combined mechanical advantage of the full arm?
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- The biceps are used in contracting the arm. The ideal mechanical advantage is the ratio of the input lever length to the output lever length, or (1.25 cm)/(45 cm) = 0.028.
- The triceps are used in extending the arm. The ideal mechanical advantage is the ratio of the input lever length to the output lever length, or (0.8 cm)/(45 cm) = 0.018.
- The mechanical advantages multiply, so the combined mechanical advantage for extending the arm is 0.15×0.018 = 0.0027.
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- A block and tackle with a mechanical advantage of 3.0 is used to pull a boat up a ramp. The ramp is 1.5 m high and 5.0 m long.
- What is the combined mechanical advantage of this arrangement?
- How much input force must be applied to obtain the needed output force of 3,500 N?
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- The combined mechanical advantage of the arrangement is 10.
- The input force required is 350 N.
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- When early humans were hunting large game, why would small mechanical advantages of their legs be advantageous?
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The lever trades off speed (or distance moved) for force. For a lever with a small mechanical advantage (less than one), more force is required to move the legs but the legs move a larger distance. In other words, small mechanical advantages allow humans to run faster—although their muscles must deliver more force to do so.
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- A bicycle’s crank has two front gears, with 39 and 52 teeth. The rear cassette has five gears, with 15, 17, 19, 22, and 25 teeth.
- What are all the possible gear ratios for the bicycle?
- What are all the possible mechanical advantages?
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Answer: - 2.60, 2.29, 2.05, 1.77, 1.56, 3.47, 3.06, 2.74, 2.36, 2.08
- 0.385, 0.436, 0.487, 0.564, 0.641, 0.288, 0.327, 0.365, 0.423, 0.481
Solution: - In each combination of front and back gears, the gear ratio is the ratio of the input teeth to the output teeth.
- The mechanical advantage is the ratio of output teeth to input teeth.
Bicycle gear ratios and mechanical advantages Front gear
(input teeth) | Back gear
(output teeth) | Gear ratio | Mechanical advantage |
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| 39 | 15 | 2.60 | 0.385 | | 39 | 17 | 2.29 | 0.436 | | 39 | 19 | 2.05 | 0.487 | | 39 | 22 | 1.77 | 0.564 | | 39 | 25 | 1.56 | 0.641 | | 52 | 15 | 3.47 | 0.288 | | 52 | 17 | 3.06 | 0.327 | | 52 | 19 | 2.74 | 0.365 | | 52 | 22 | 2.36 | 0.423 | | 52 | 25 | 2.08 | 0.481 |
- Nadja has the multispeed bicycle described in the last problem and is stopped at a traffic light.
- When the light turns green, what mechanical advantage should she use at first, low or high? What gear ratio, low or high? Which gears should she use? Why?
- As she goes faster and faster, what gear ratio and mechanical advantage should she use, low or high? Which gears should she use? Why?
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- When she is starting, she is moving slowly and will want to move the wheels with the maximum output force. She wants to maximize force, not speed. That means that she wants a high mechanical advantage, which corresponds to a low gear ratio. The combination of front gear 39 (small chainring) and back gear 25 (largest gear) provides the highest mechanical advantage.
- When she is moving faster, Nadja wants to pedal at a comfortable rate while spinning the back wheel around very quickly. She needs to maximize speed, not force. This corresponds to a high gear ratio or low mechanical advantage. The combination of front gear 52 (large chainring) and back gear 15 (smallest gear) provides the highest gear ratio and lowest mechanical advantage.
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