Understanding Acceleration and Center of Mass in Shock Absorption

• ThEmptyTree
In summary, the person is suffering a shock because they are reaching their maximum velocity and then losing momentum. The forces acting on the person are gravity and the normal to the ground.
ThEmptyTree
Homework Statement
The compressive force per area necessary to break the tibia in the lower leg is about ##F/A = 1.6 × 108 N/m^2##. The smallest cross sectional area of the tibia, about 3.2 cm2, is slightly above the ankle. Suppose a person of mass ##m = 60 kg## jumps to the ground from a height ##h_0 = 2.0 m## and absorbs the shock of hitting the ground by bending the knees. Assume that there is constant deceleration during the collision with the ground, and that the person lowers their center of mass by an amount ##d = 1.0 cm## from the time they hit the ground until they stop moving.

(a) What is the collision time ##\Delta{t_{col}}##, to 2 significant figures?
(b) Find ##N_{ave}##, the magnitude of the average force exerted by the ground on the person
during the collision in Newtons.
(c) What is the ratio of the average force of the ground on the person to the gravitational force on the person? Can we effectively ignore the gravitational force during the collision?
(d) Will the person break his ankle?
Relevant Equations
Newton's 2nd Law: ##\overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}##
I don't attempt solving a problem until I fully understand it, conceptually.

After the hit (when maximum velocity is reached) the person starts losing momentum, having a constant upwards acceleration. The forces acting on the person are gravity and the normal to the ground.
$$N - mg = ma$$
##N>mg## and that's why the person suffers the shock.
My question is, how does the person lower the center of mass of its body, if the acceleration of the center of mass is
$$\overrightarrow{A_{cm}}=\frac{\overrightarrow{F_{ext}}}{M}$$
If the acceleration is upwards, shouldn't (hypothetically) the center of mass go in the direction of the acceleration?

One explanation that came to my mind was the fact that if we consider the body a reference frame which accelerates upwards, then the fictitious force would be downwards, but it would only apply to this case.

I am very confused, can someone explain this to me please?

What would happen if you applied equal and opposite forces at either end of an object?

Consider its acceleration and compression force.

PeroK said:
What would happen if you applied equal and opposite forces at either end of an object?

Consider its acceleration and compression force.
I don't quite get it. I try to compare it to applying forces at both ends of a rope but I'm bad at visualizing things. I have never worked with compression forces and it hasn't been introduced in the course yet. Aren't the only external forces gravity and normal?

However, introducing the idea of compressing force might explain why the person has acceleration despite not moving, like an object being compressed. In this case the compression force is determined by the person crouching (so it's from the top)?

I have been meditating on what you said for half an hour, but no results. Further hints will be appreciated. Thanks a lot.

Newton's second law describes motion based on the net external force, not total compression forces.

ThEmptyTree said:
If the acceleration is upwards, shouldn't (hypothetically) the center of mass go in the direction of the acceleration?
Don't confuse direction of motion with direction of acceleration. If you throw a stone up, even while it is rising the acceleration is downwards.

jim mcnamara, ThEmptyTree and Steve4Physics
haruspex said:
Don't confuse direction of motion with direction of acceleration. If you throw a stone up, even while it is rising the acceleration is downwards.
Yup, I think that's it. The speed is still downwards. Thanks.

1. What is compressive strength of bones?

Compressive strength of bones refers to the ability of bones to withstand a compressive force without breaking or fracturing. It is a measure of the maximum stress a bone can handle before it fails.

2. How is compressive strength of bones measured?

Compressive strength of bones is typically measured using a compression test, where a bone sample is placed between two plates and a compressive force is applied until the bone breaks. The maximum force applied is then divided by the cross-sectional area of the bone to determine its compressive strength.

3. What factors affect the compressive strength of bones?

The compressive strength of bones can be affected by various factors such as age, gender, bone density, and overall health. Bones tend to have higher compressive strength in younger individuals and in those with higher bone density. Certain medical conditions and lifestyle factors can also impact bone strength.

4. Why is it important to study the compressive strength of bones?

Studying the compressive strength of bones is important for understanding bone health and potential risks for fractures. It can also help in the development of treatments and prevention strategies for bone-related conditions such as osteoporosis.

5. Can the compressive strength of bones be improved?

Yes, the compressive strength of bones can be improved through regular weight-bearing exercise, proper nutrition, and avoiding risk factors such as smoking and excessive alcohol consumption. In some cases, medical treatments may also be recommended to improve bone strength.

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