The physics of walking and running
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- The physics of walking and running
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The physics of walking and running
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[ID] => 556878
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[post_date] => 2025-01-09 07:39:52
[post_date_gmt] => 2025-01-09 12:39:52
[post_content] => Practice Passage (Question 1-5)
*This passage is the property of Khan Academy and has been reformatted into an AAMC-style interface in their entirety by MedLife Mastery. MedLife Mastery does not endorse and is not an affiliate of Khan Academy.
Walking is energy efficient. In a walking human, one leg swings forward while the other leg’s foot stays planted on the ground. When walking at natural speed, the swinging leg uses muscle force to move forward and immediately relaxes, allowing the force of gravity to move it to the ground. Simultaneously, the planted leg moves forward with largely passive rotation at the hip. The plant leg only needs to stay straight and the swinging leg’s knee only slightly bends to allow it to pass underneath the body.
The swinging leg can be modeled as a physical pendulum: a thin uniform rod of mass m rotating about a point a distance r from its center of mass. Swinging freely under gravitational acceleration, g, such a physical pendulum with moment of inertia I will swing with a period T, as shown in Equation 1.
For a uniform, thin rod of length l with a pivoted end, 
The natural walking step length is roughly 
. The natural speed of walking, v, is the step length divided by the time required to take the step.
To move faster or slower than the natural speed, the legs do not move at their natural frequencies or with the natural step length. Instead, the muscles produce forces (hence torques) to move the body forward. The maximum force a muscle can produce, Fₘₐₓ, is proportional to its cross sectional area, A, which is proportional to the square of the length: Fₘₐₓ∝A∝l². The maximum torque that the muscle can exert about its pivot point, Lₘₐₓ, is proportional to the product of Fₘₐₓ and its length: Lₘₐₓ∝Fₘₐₓl. The mass of the leg is proportional to the mass of the muscle, which is proportional to the product of the area and the length. Unlike in the case of walking, the period of a pendulum acted on by maximum torque Lₘₐₓ is (constant of proportionality not organism-specific) given by Equation 2.
[post_title] => The physics of walking and running
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[question] => Modeling the leg as a uniform cylindrical rod, which of the following changes would most increase the moment of inertia of a runner’s leg?
[value] => Array
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[answer] => 1
[description] => Reason for the Correct Answer:
The mass of a uniform cylinder is m = ρπR²l, where is the material density and R is the radius.
That means that we can form an expression for the moment of
A 10% increase in ρ, R, or I would multiply the I above by 1.1, 1.1² = 1.210, or 1.1³ = 1.3310, respectively. A 10% increase in the length of the runner’s leg, then, would most increase the leg’s moment of inertia.
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[answers] => Array
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[0] => Array
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[each_answer] => A. A 10% increase in the leg’s length
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[1] => Array
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[each_answer] => B. A 10% increase in the leg’s diameter
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[2] => Array
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[each_answer] => C. A 10% increase in the leg’s density
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[3] => Array
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[each_answer] => D. All of these changes would have equal effect
)
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[1] => Array
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[quiz_unique_key] => 3873426850
[question] => Suppose that a 2m tall person is unable to bend one leg (or swing it to the side), which is 1m long, but instead had to use their calf muscles to raise their heel 5cm above the ball of their foot to let the stiffened leg swing underneath them. The energy that the calf muscles use to perform this up-down motion is all lost as heat. What is the minimum amount of extra energy they would have to expend to walk 1km? (Assume that natural step lengths and periods are unaffected.)
[value] => Array
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[answer] => 4
[description] => Reason for the Correct Answer:
The person would have to take 5,000 steps to go 1km:
Half of these steps would waste energy.
The amount of energy wasted would be the additional potential energy from lifting the body up with the calf muscles that they cannot get back,
PE = mgh = mg x 5cm.
You have not been provided with the person’s mass, m, so you have insufficient information to solve the problem.
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[answers] => Array
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[0] => Array
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[each_answer] => A. 50,000 J
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[1] => Array
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[each_answer] => B. 125,000 J
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[2] => Array
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[each_answer] => C. 250,000 J
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[3] => Array
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[each_answer] => D. Insufficient information
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[2] => Array
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[quiz_unique_key] => 83407773
[question] => These dynamics apply not just to humans, but other animals as well. Suppose that a panda and a giraffe were both moving at their natural walking speeds. How much faster would the giraffe move than the panda if the giraffe’s leg length were 2m and mass were 1,600 kg, while the same measurements for the panda were 50cm and 75kg?
[value] => Array
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[answer] => 1
[description] => Reason for the Correct Answer:
The center of mass of a thin uniform rod of length l is located in the middle, so r in (Equation 1) is 
Plug the expression for I into (Equation 1):
Note that T is the period of a pendulum, which is the round trip. The time to take a step is 
The natural step length is roughly 
.
The natural walking speed for each animal is 
Plugging in for each animal’s leg length and taking the ratio:
, hence the giraffe walks 2 times as quickly as the panda.
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[answers] => Array
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[0] => Array
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[each_answer] => A. The giraffe would walk 2 times as quickly as the panda.
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[1] => Array
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[each_answer] => B. The giraffe would walk 4 times as quickly as the panda.
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[2] => Array
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[each_answer] => C. The giraffe would walk 1.4 times as quickly as the panda.
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[each_answer] => D. They would walk at the same speed.
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[quiz_unique_key] => 2261298308
[question] => If both a giraffe and a panda were to briefly run at their maximum speeds without becoming exhausted, which animal would have a natural speed advantage based on the models presented in the passage? (Assume that stride lengths are still proportional to leg length. Take giraffe’s leg length were 2m and mass were 1,600 kg, while the same measurements for the panda were 50cm and 75kg.)
[value] => Array
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[answer] => 2
[description] => Reason for the Correct Answer:
The mass of the leg is proportional to the product of the area and the length:
Plugging the mass into the expression for the moment of inertia (which is still valid in the case of running),
Plug this expression for I into (Equation 2), bearing in mind that 
we find that 
Each animal’s maximum running speed is then 
. As none of the constants of proportionality were specific to either of the animals, both would run at the same maximum speed! (One of the reasons that some animals like cheetahs are so much faster than other animals is that they use abdominal muscles in a way that these models don’t take into account.)
)
[answers] => Array
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[0] => Array
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[each_answer] => A. The giraffe would run 1.4 times as quickly as the panda.
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[1] => Array
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[each_answer] => B. They would run at the same speed.
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[2] => Array
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[each_answer] => C. The giraffe would run 2 times as quickly as the panda.
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[3] => Array
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[each_answer] => D. The panda would run 1.4 times as quickly as the giraffe.
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[quiz_unique_key] => 2261298308
[question] => In order to run efficiently, a runner will let their arms swing so that the right arm is going backward as the right leg goes forward, vice versa, and similarly with the left arm and leg. As they run faster, the runner increases their stride rate. How should a runner change their arm motion to run most efficiently?
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[answer] => 1
[description] => Reason for the Correct Answer:
To match the increased stride rate, the arms must swing back and forth more.
To do that most efficiently, it is best to let gravity do the work instead of using muscles.
Use (Equation 1) and the fact that shorter, more compact objects tend to have lower moments of inertia.
To shorten Tₐᵣₘ, the runner should shorten their arms. The only way to shorten their arms (without a hacksaw or surgical tools!) is to bend their elbows more.
)
[answers] => Array
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[0] => Array
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[each_answer] => A. bend their elbows more
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[each_answer] => B. straighten their arms
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[each_answer] => C. keep their arms the same length
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[each_answer] => D. arm length does not affect running efficiency
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