Centripetal force and friction

Question

Here is the question I need help on:

A hemispherical pile of snow with a height of 20 m is piled up. An extreme skier starts at the top of this snowball, and she skies down finding that everytime she tries it she flies off at the same angle α with respect to a vertical line from the center of the hemisphere of snow to the top. Neglecting friction between skies and snow, estimate this angle. How would the angle change if you include friction?

By my understanding, the velocity is perpendicular to the "string" tension which pulls it towards the center. However, the skier isn't really "tied" to the center here so I'm not sure at what point you'd say the skier breaks from the hemisphere and goes off at the angle. I'm assuming it is 90 degrees at to the radius at the point of breaking but without knowing when the skier flies off, I'm not sure how to find the angle with respect to the vertical line. Any help? Hopefully I'm not on the wrong path completely.

Answer 1

In accordance with the homework policy, here are some things to think about in order to solve this.

There is a force holding the skier attached to the snow globe: that is gravity. As the angle gets steeper, the component of gravity pointing towards the center of the sphere gets smaller. This force needs to be strong enough to keep the skier moving in a circular path.

There comes a point where that is not the case. You know the velocity at a given angle (from angle you calculate height, and from height, potential energy lost = kinetic energy gained). Thus you know what the force needs to be ($F=\frac{v^2}{r}$). And the component of gravity pointing inwards is also a function of angle.

Put it all together and solve. Obviously, if you include friction, the kinetic energy gained will be less than the potential energy lost: this means the skier will go more .... and stay attached to the globe for ....

Answer 2

As skier slides down, the normal force between skies and dome reduces. It is maximum at top but reduces further as he slides down. It is the normal force which is keeping skier adhered to dome.

As the speed of skier increases, there is an increase in tangential speed which gives an increased centrifugal force.

Centripetal force tends to skier adhered provided by normal force $mg sin\theta$ ($\theta$ is angle wrt horizontal ground) and centrifugal force provided by tangential speed $mv^2/r$ tends to fly off skies. The point at which normal force equals centrifugal force would be the point where skier leaves hemisphere.

You can easily calculate tangential speed using conservation of energy.

Equating $mv^2/r$ to $mg sin\theta$ gives required h.

Without friction ,it will escape bowl at $h= \frac23 R$. This will help you to take out angle.

Including friction would use loss of energy due to friction .

This will reduce v and hence angle at which skier leaves.

Answer 3

It's a pretty easy one!! As per the homework policy of this community, I won't tell you the exact answer but I'll tell you the method you need to use.

See, it's the combination of the NORMAL force and GRAVITY which provides the skier with the required centripetal force.

Also, as the skier's height decreases (PE decreases) , but KE increases (by energy conservation) which is why the skier's velocity increases.

So, you'll get 2 equations:

1) From Energy conservation

2) From circular motion equation (At angle alpha, Normal will be 0 as skier losses contact, mg provides all the centripetal force.)

You may then solve for alpha.

Now, if there is friction, your first equation will change a bit as a term containing 'work done by friction' will get added. If there is friction, the angle alpha will increase.

I hope, these many hints should be enough for you.