Skier on a snowball (Newtonian Mechanics)

In summary, the skier will come off the surface at an angle of 60°, using the principles of dynamics and the equations E = KE + U and KE = (1/2)mv^2, with the assumption that the gravitational force and the force of motion are the only forces acting on the skier at the point of departure. The skier's acceleration in the radial direction (centripetal acceleration) can be found by setting the centripetal force (gravity) equal to m*a_c and solving for a_c, which is (v^2)/R. The resulting angle of 60° can then be plugged into the energy equation to find the final kinetic and potential energies.
  • #1
derravaragh
24
0

Homework Statement


A skier slides down a giant snowball (= sphere of radius R) with negligible friction. H starts at the top with very small velocity. Determine the angle θ_f where the skier will come off the surface. Use these principles of dynamics: (1) Energy is a constant of the motion. (2) The normal force of the contact decreases as the skier descends, and N = 0 at the point where the skier comes off the surface. (3) a = R*alpha*t(hat) - R*(w^2)*r(hat) where a is the acceleration vector, t(hat) is the time vector, and r(hat) is the position vector.


Homework Equations


E = KE + U where E is energy, KE is kinetic energy, and U is potential energy (I understand these may not be the standard)
KE = (1/2)mv^2



The Attempt at a Solution


For this problem, I am at a loss for what to do. I've worked it out a few times, taking into account all the factors. Using the first principle, I determined the initial kinetic energy is 0 as the initial velocity is so small, it's negligible, and the initial potential energy is merely 2Rmg. Next, I determined the final kinetic energy to be (1/2)mv^2 and final potential to be mg(R+Rcosθ_f).
Therefore, the energy equation is: 2Rmg = (1/2)mv^2 + mg(R+Rcosθ_f)

Next, I considered the fact that the centripetal force = the normal force, therefore at the point when the skier leaves the surface, the acceleration is only reliant on the tangential acceleration.

This is where I get confused. I can't seem to figure out how to make the connection between the acceleration and velocity for this problem, and be able to solve for θ_f.

Where should I go from here, assuming I'm even on the right track.
 
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  • #2
derravaragh said:
the centripetal force = the normal force
That's wrong. You were given, correctly, that at point of departure the normal force will be zero. What are all the forces acting on the skier at this point?
 
  • #3
The gravitational force, which I took to be just F_g = -mg at this point because there is nothing below him until he reaches the ground. The only other force I can think of would be the force of his motion.
 
  • #4
derravaragh said:
The gravitational force, which I took to be just F_g = -mg at this point because there is nothing below him until he reaches the ground. The only other force I can think of would be the force of his motion.
Right (but motion is not a force... you may be thinking of inertia, which is momentum).
What is the skier's acceleration in the radial direction (you previously named this)? Since, a microsecond before losing contact, gravity is the only force available to provide that acceleration, what equation can you write?
 
  • #5
Ok, so the radial acceleration is what I called centripetal acceleration, so (and this may be wrong way of looking at it, but I'm going for the end result) mg = m*a_c where a_c = (v^2)/R therefore g = (v^2)/R and (v^2) = Rg which I can plug into an equation I came to before to obtain an angle of 60°.
 
  • #6
derravaragh said:
Ok, so the radial acceleration is what I called centripetal acceleration, so (and this may be wrong way of looking at it, but I'm going for the end result) mg = m*a_c where a_c = (v^2)/R therefore g = (v^2)/R and (v^2) = Rg which I can plug into an equation I came to before to obtain an angle of 60°.
Right idea, but notice that g is not in quite the right direction. What component of g is?
 

FAQ: Skier on a snowball (Newtonian Mechanics)

1. How does a skier on a snowball demonstrate Newtonian mechanics?

The concept of Newtonian mechanics is based on Newton's three laws of motion. A skier on a snowball demonstrates these laws in action. The first law, also known as the law of inertia, states that an object at rest will remain at rest unless acted upon by an external force. In this case, the skier remains on the snowball due to the force of friction between the snow and the skier's skis. The second law states that the acceleration of an object is directly proportional to the force applied to it and inversely proportional to its mass. As the skier leans forward, the force of gravity pulls them down, causing the snowball to accelerate. The third law states that every action has an equal and opposite reaction. As the skier pushes against the snow with their skis, the snow exerts an equal and opposite force, propelling the snowball forward.

2. What factors affect the speed and direction of the skier on the snowball?

The speed and direction of the skier on the snowball are affected by several factors. These include the mass of the snowball and the skier, the force of gravity, the angle of the slope, and the coefficient of friction between the snow and the skier's skis. Additionally, the skier's technique and movements, such as leaning forward or turning, can also impact their speed and direction on the snowball.

3. How does friction play a role in a skier's movement on the snowball?

Friction is crucial in a skier's movement on the snowball. Friction is the force that opposes motion between two surfaces in contact. In this case, the friction between the snow and the skier's skis allows the skier to maintain their balance on the snowball. Without friction, the skier would not be able to stay on the snowball and would likely fall off.

4. Can a skier on a snowball ever reach a constant speed or will they eventually stop?

According to Newton's first law of motion, an object in motion will remain at a constant speed unless acted upon by an external force. Therefore, if the snowball and the skier are on a horizontal surface with no external forces acting on them, they will eventually reach a constant speed and continue moving at that speed indefinitely. However, on a sloped surface, the force of gravity will cause the snowball and skier to accelerate, and they will not reach a constant speed unless the slope becomes less steep.

5. How is the skier's center of mass related to their movement on the snowball?

The center of mass is the point where an object's mass is evenly distributed. In the case of a skier on a snowball, their center of mass plays a crucial role in their movement. As the skier shifts their weight and changes their center of mass, they can control their speed and direction on the snowball. For example, leaning forward will cause the snowball and skier to accelerate, while leaning backward will slow them down. Additionally, the skier's center of mass must remain within the base of support (the area covered by the snowball and the skier's skis) to maintain their balance and prevent them from falling off the snowball.

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