Skier on a hemispherical mountain

  • Thread starter amcca064
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In summary, the conversation discusses a skier of mass 60kg sliding down from the top of a frictionless icy hemispherical mountain with a radius of 100m. The conversation continues with a request to draw a free body diagram and write Newton's equations at a specific moment, and then find the angle 'alpha' at which the skier will lose contact with the mountain surface. The free body diagram includes an x-axis tangent to the mountain and a y-axis along the radius, with the angle alpha measured from the origin of the circle. The Newton's equations at this point are Fx = mgcos(alpha) and Fy = n - mgsin(alpha). The conversation ends with a request for help in linking momentum to
  • #1
amcca064
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The question is:

A skier of mass 60kg and initially at rest, slides down from the top fo a frictionless icy hemispherical mountain with a radius of 100m.

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

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

Ok so I have my free body diagram set up with the x-axis being tangent to the mountain, and the y-axis along the radius. The angle alpha is measured from the origin of the circle (mountain) which makes the component of the gravitational force mgcos(alpha) tangent to the circle along the direction of motion (to the right from the top of the mountain in my case). The component mgsin(alpha) is along the radius. The Newtons equations in this case are Fx = mgcos(alpha) and Fy = n - mgsin(alpha). This is as far as I've gotten, I think part b has to do with momentum but I cannot think of a way to link this to mg to solve the problem! Any help would be much appreciated. Thanks

Aidan
 
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  • #2
double post sorry
 
  • #3
,

Thank you for providing your free body diagram and Newton's equations. You are correct that the angle alpha is important in determining when the skier will lose contact with the mountain surface. To solve for this angle, we can use the concept of centripetal force.

As the skier slides down the mountain, they are experiencing a centripetal force directed towards the center of the circle (mountain). This force is equal to the skier's mass times their tangential velocity squared divided by the radius of the circle (Fc = m(v^2)/r). At the point where the skier loses contact with the mountain, the normal force (n) becomes zero, meaning that the only force acting on the skier is their weight (mg). This weight must then be equal to the centripetal force, which we can express as mg = m(v^2)/r. We can rearrange this equation to solve for the tangential velocity (v) at the point of losing contact, which is v = √(gr).

Now, we can use trigonometry to find the angle alpha. We know that the tangential velocity (v) is equal to the component of the gravitational force mgcos(alpha) in the x-direction, so we can set these two equal to each other: v = mgcos(alpha). Plugging in the value we just found for v, we get √(gr) = mgcos(alpha). We can then solve for alpha by taking the inverse cosine of both sides: alpha = cos^-1(√(gr)/mg). This gives us the angle at which the skier will lose contact with the mountain surface.

I hope this helps! Let me know if you have any further questions.
 

Related to Skier on a hemispherical mountain

1. How does the shape of the mountain affect the skier's experience?

The shape of the mountain greatly affects the skier's experience. A hemispherical mountain has a curved surface, which means the slope changes constantly and can be more challenging for skiers. It also affects the speed and direction of the skier's movements.

2. How do skiers maintain their balance on a hemispherical mountain?

Skiers maintain their balance on a hemispherical mountain by constantly adjusting their body position and weight distribution. They also use their poles and edges of their skis to control their movements and maintain stability.

3. Is skiing on a hemispherical mountain more dangerous than on a regular mountain?

It can be argued that skiing on a hemispherical mountain is more dangerous due to the constantly changing slope and potential for unexpected turns or drops. However, with proper safety precautions and training, it can be a thrilling and safe experience.

4. Are there any advantages to skiing on a hemispherical mountain?

One advantage of skiing on a hemispherical mountain is the variety of terrain it offers. Skiers can experience a range of slopes and challenges in one run. It also allows for a more dynamic and exciting skiing experience.

5. What factors contribute to the difficulty of skiing on a hemispherical mountain?

The difficulty of skiing on a hemispherical mountain depends on several factors, including the steepness and shape of the slope, the snow conditions, and the skier's skill level. It also requires a higher level of physical and mental agility due to the constant changes in terrain.

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