Solving a Skiing Problem: Finding the Angle of Loss of Contact

• amcca064
In summary, the 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. At some point below the top of the mountain, the skier will reach a speed where the component of gravity can no longer keep him on the slope, causing him to lose contact with the mountain surface. This can be calculated using the equations Fx = mgcosalpha and Fy = n - mgsinalpha. The angle 'alpha' at which the skier will lose contact with the mountain surface can be found by setting the normal force to zero in the second equation, resulting
amcca064
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

As the skier goes down the slope his speed increases. Also the component of gravity that must keep him on the slope decreases. At some stage his speed becomes such that the component of gravity cannot keep him on the slope anymore and he flies off. At this point he will loose contact with the slope and the normal force will be zero. Which means that your second equation will become

$$\Sigma F_y = -w\sin(\alpha)$$

where w is the weight of the skier. In the x-direction we need to consider the tangential acceleration

$$\Sigma F_x = m a_t = m\alpha r$$

where $$\alpha$$ is the angular acceleration of the skier

Hi Aidan,

You are on the right track with your free body diagram and equations. To find the angle at which the skier will lose contact with the mountain, we need to consider the forces acting on the skier and the motion of the skier.

At the moment when the skier loses contact with the mountain, the normal force will become zero. This means that the gravitational force and the centripetal force (acting towards the centre of the semicircle) will be equal in magnitude. We can set up an equation using the centripetal force formula, Fc = mv^2/R, where v is the velocity of the skier and R is the radius of the semicircle. We can also use the equation for the gravitational force, Fg = mgcos(alpha), to substitute for the centripetal force. This gives us:

mgcos(alpha) = mv^2/R

We can rearrange this equation to solve for the angle alpha:

alpha = cos^-1(v^2/(gR))

To find the velocity of the skier, we can use the equations of motion, specifically the equation for velocity in the x-direction, vx = v0 + axt, where v0 is the initial velocity (in this case, 0 since the skier starts at rest), ax is the acceleration in the x-direction (which is equal to the centripetal acceleration, v^2/R), and t is the time taken to reach the point where the skier loses contact with the mountain. We can also use the equation for displacement in the x-direction, x = x0 + v0t + (1/2)axt^2, where x0 is the initial position (which is also 0 in this case). We know that the displacement in the x-direction is equal to the radius of the semicircle, R, so we can substitute and solve for t. This gives us:

t = sqrt(2R/v^2)

Now we can substitute this value for t into the equation for velocity in the x-direction, and then substitute that into our equation for alpha. This gives us:

alpha = cos^-1((v^2/(gR))(2R/v^2))

Simplifying, we get:

alpha = cos^-1(2/g)

This means that the angle at which the skier will lose contact with the mountain is equal to the inverse cosine of 2/g,

What is the purpose of finding the angle of loss of contact in skiing?

The angle of loss of contact is used to determine the optimal angle at which skiers should lean while skiing in order to maintain their balance and control their speed. It is an important factor in improving skiing technique and preventing accidents.

How is the angle of loss of contact calculated?

The angle of loss of contact is typically calculated using trigonometric functions and the skier's weight distribution on their skis. It can also be measured using specialized equipment such as an inclinometer.

What factors can affect the angle of loss of contact?

The angle of loss of contact can be affected by a variety of factors, including the slope of the terrain, the type and condition of the snow, the skier's speed and technique, and their equipment (such as ski length and stiffness).

Why is it important to accurately measure the angle of loss of contact?

An accurate measurement of the angle of loss of contact allows skiers to adjust their technique and position on the skis to maintain control and prevent accidents. It can also help coaches and instructors provide targeted feedback to improve a skier's performance.

What are some techniques for improving the angle of loss of contact?

Some techniques for improving the angle of loss of contact include maintaining a strong and balanced stance, keeping your weight centered over your skis, and adjusting your body position and weight distribution as needed for different terrain and conditions. Working with a ski instructor or coach can also help improve technique and optimize the angle of loss of contact.

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