Ok, back to the turning skater. Here is a force diagram including the fake force.
So, just 4 forces. For an object in equilibrium (which would be true in our non-inertial reference frame), the following must be true:
The net force in both the x- and y-directions must be zero as well as the torque (which we can treat as a scalar here) about some point. Here, I will use the scalar definition of torque about some point o:
Where F is the applied force, r is the distance from the force to the point "o" and θ is the angle between F and r. If you want to know more about torque, this older post might be useful.
Oh, one more thing. What about the fake force? This depends on both the mass of the skater and the acceleration of the frame (which is also the skater). Since the skater is moving in a circular motion, the magnitude of this fake force would be:
Here v is the magnitude of the skater's velocity and R is the radius of the circle the skater is moving in. Now, there is a trick here. If the skater is leaning in towards the center of the circle, different part's of the body will be at different distances from the center. If the radius is large enough, these distance differences won't really matter. For the rest of the calculations, I will assume this fake force acts at the center of mass of the person.
Now I can start to put in some values. If we look at the total torque about the point where the skates touch the ice, I can ignore both the normal and frictional forces since they produce no torque. Oh, let me assume that the center of mass is in the middle of a skater with a height of h. This gives:
This says a few things:
- The faster you go, the more the skater leans (smaller angle).
- Skater mass doesn't matter.
- The height of the skater doesn't mater either.
It seems you're delving into the physics of a turning skater! Let's break this down systematically.
Firstly, you're discussing the dynamics of a skater's motion, analyzing forces involved and their impact on equilibrium in a non-inertial reference frame. The reference to the "fake force" likely alludes to a fictitious force, like the centrifugal force experienced in a rotating frame of reference.
To address the concepts used in your discussion:
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Force Diagram and Equilibrium:
- In an equilibrium state, both the net forces in the x- and y-directions must be zero.
- The torque around a point is defined as the product of the applied force, the distance from the force to the point, and the angle between them.
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Torque:
- Torque (τ) about a point "o" is given by τ = r F sin(θ), where r is the distance, F is the force applied, and θ is the angle between the force and the position vector.
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Fake Force in Circular Motion:
- In circular motion, a fictitious or fake force can arise due to the acceleration of the frame (the skater). Its magnitude can be expressed as F_fake = m * v^2 / R, where m is the mass, v is the velocity, and R is the radius of the circular path.
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Dependence of Fake Force:
- The magnitude of the fake force depends on the velocity of the skater and the radius of the circular path. When the skater leans towards the center, different body parts are at different distances from the center. However, if the radius is sufficiently large, these differences become negligible.
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Impact of Parameters:
- The calculations suggest that the skater's mass and height do not directly influence the total torque about the point where the skates touch the ice. Instead, the skater's speed affects the angle of lean.
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Relation Between Speed and Lean:
- As the skater accelerates (increasing speed), the angle of lean decreases. This means that at higher speeds, the skater leans less towards the center.
Understanding these principles enables an analysis of the forces acting on a turning skater, considering equilibrium, torques, and the dynamics of circular motion. The relationship between speed and lean angle seems to be a crucial aspect of this analysis.
