Skier Challenge: Analyzing Newton's Equations for Alpha

In summary, we discussed a problem involving a skier of mass 60kg sliding down a frictionless, icy, hemispherical mountain with a radius of curvature R of 100m. We drew a free body diagram and wrote Newton's equations for the moment when the skier is at some point below the top of the mountain. We also discussed finding the angle 'alpha' with the horizontal surface at which the skier will lose contact with the mountain surface. We then used energy conservation to solve for the velocity and potential energy at different positions of the skier.
  • #1
amcca064
32
0
Ok so the question is,
"A skier of mass 60kg, initially at rest, slides down from the top of a frictionless, icy, hemispherical, mountain with a radius of curvature R of 100m.

a)Draw a free body diagram and write the Newton's Equations at the moment when he/she is at some point below the top of the mountain.

b)Find the angle 'alpha' with the horizontal surface at which he/she will lose contact with the mountain surface."

Ok so for this question the free body diagram I have is set up so that the angle is measured from the centre of the hemispherical mountain to the skier. In this way, the x axis' is tangent to the semicircle, and the y-axis follows the radius of the semicircle. The skier in my diagram is moving to the right along the mountain, which means that the component of the gravitational force that is acting on him/her in the x direction is
mgcos(alpha) and the component along the y direction is mgsin(alpha). The normal force is along y axis. The Newton's equations are

Fx = mgcosalpha
and
Fy = n - mgsinalpha

So far I think I have that right, now the difficult part is part b. I don't know what to look for in this part! What equations should I use and how should I set them up? I think this question has to do with momentum, i.e. when does her horizontal momentum carry her off the mountain, but I don't know how to set the equations up to get there. Any help would be greatly appreciated. Thank you.
Aidan
 
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  • #2
Hint 1: didn't you forget about a force? (Circular motion!)
Hint 2: if contact is lost, which force must equal zero?
 
  • #3
Even if its circular motion, that means that Fy= Fr which is still = n - mgsinalpha
And when contact is lost, the normal force will be zero, which means that Fr = -mgsinalpha
 
Last edited:
  • #4
amcca064 said:
Even if its circular motion, that means that Fy= Fr which is still = n - mgsinalpha
And when contact is lost, the normal force will be zero, which means that Fr = -mgsinalpha

Ok, I assumed the skier is skiing down 'from left to right', but this doesn't matter. You have: N = mg*sin(alpha) - m*v^2 / R. The only thing you have to find is the velocity v the skier has when the line connecting the center of curvature and the skier's position makes some angle alpha with the horizontal. You can use energy conservation.
 
  • #5
ok... so what you said is basically, mgsinalpha=m[(v^2) / r] ----> gsinalpha = (v^2)/r ----> (9.81)(100)sinalpha=v^2 so sinalpha= v^2/981
which is two unknowns and one equation. I know I need to use energy conservation to find the answer, I just don't know how...
 
  • #6
amcca064 said:
...
which is two unknowns and one equation. I know I need to use energy conservation to find the answer, I just don't know how...

Select two positions of the skier so that one is at the top and the other is at some angle alpha. What is the potential energy of the skier at the top? What about the kinetic energy at the top? Further on, what is the potential energy of the skier at some point alpha? ANd what about the kinetic energy? Now use energy conservation. The sum of kinetic and potential energy of the skier at the top must equal the sum of potential and kinetic energy of the skier at some angle alpha.
 
  • #7
Ki + Ui = Kf +Uf makes things a lot easier now, thanks for bringing that to my attention, it was the one thing I couldn't piece together. thanks
 
  • #8
I have two questions about the conservation of energy part.

First: Is the v in the final kinetic energy left as v and solved for?
Second: How can you determine the height, h, in the final potential energy, Uf?
 

1. What is the purpose of the "Skier Challenge" experiment?

The purpose of the "Skier Challenge" experiment is to analyze and apply Newton's equations of motion to the motion of a skier on an inclined slope, in order to determine the skier's acceleration, velocity, and position at different points along the slope.

2. What are Newton's equations of motion?

Newton's equations of motion, also known as the laws of motion, are three fundamental principles that describe the relationship between the forces acting on an object and its resulting motion. These are:
- The first law, also known as the law of inertia, states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
- The second law, also known as the law of acceleration, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
- The third law, also known as the law of action and reaction, states that for every action, there is an equal and opposite reaction.

3. How do you apply Newton's equations to the "Skier Challenge" experiment?

In the "Skier Challenge" experiment, Newton's equations are applied by considering the forces acting on the skier, such as gravity, normal force, and friction, and using them to calculate the skier's acceleration, velocity, and position at different points along the slope. This involves breaking down the forces into their components in the direction of motion and using the relevant equation to solve for the unknown variables.

4. What is the significance of the variable "Alpha" in the experiment?

The variable "Alpha" represents the angle of inclination of the slope in the "Skier Challenge" experiment. It is an important factor in determining the forces acting on the skier and ultimately, their motion down the slope. By varying the value of "Alpha", we can observe how the skier's acceleration, velocity, and position change accordingly, and gain a better understanding of the relationship between angle of inclination and motion on an inclined slope.

5. What are some real-world applications of understanding Newton's equations in the context of the "Skier Challenge" experiment?

Understanding and applying Newton's equations in the "Skier Challenge" experiment can have various real-world applications. For instance, it can help us design safer ski slopes by considering the forces acting on skiers and minimizing potential risks. It can also be applied in the field of sports, such as in analyzing the motion of athletes in events like skiing, snowboarding, and skateboarding. Additionally, it can be useful in engineering and physics, where the principles of motion are essential in designing structures and machines that can withstand external forces.

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