A 63.7 kg skier encounters a dip in the snow's surface that has a circular cross section with radius of curvature of $r$ = 11.5 m. If the skier's speed at point A in the figure below is 7.87 m/s, what is the normal force exerted by the snow on the skier at point B?

Length from edge of depression to point above middle of the depression is 1.86 m.

Why did the skier bring his measuring tape to the slope? Because he wanted to measure up to the challenge! But don't worry, I'll crunch the numbers for you.

To find the normal force exerted by the snow on the skier at point B, we need to consider the centripetal force acting on the skier at that point. The centripetal force is provided by the normal force, so we can equate the two.

The centripetal force is given by the formula Fc = (m * v^2) / r, where m is the skier's mass, v is the skier's speed, and r is the radius of curvature of the dip.

Plugging in the given values, we have:
Fc = (63.7 kg * (7.87 m/s)^2) / 11.5 m

Now, the normal force equals the weight of the skier (mg) minus the centripetal force:
N = mg - Fc

Since the skier is not accelerating vertically, the net force in the vertical direction is zero, and the normal force balances the weight. Therefore, we can simplify the equation:
N = mg - Fc = mg - (m * v^2) / r

Plugging in the values:
N = (63.7 kg * 9.8 m/s^2) - (63.7 kg * (7.87 m/s)^2) / 11.5 m

Now, let me do the math and calculate the answer for you. *Calculating*

To find the normal force exerted by the snow on the skier at point B, we can use the concept of centripetal force.

1. First, let's calculate the acceleration of the skier at point A using the speed and radius of curvature. The acceleration can be found using the formula:

acceleration = (velocity^2) / radius

Substituting the given values, we get:

acceleration = (7.87 m/s)^2 / 11.5 m = 5.41 m/s^2

2. Next, we need to find the vertical acceleration at point B. As the skier moves in a circular path, there will be a normal force exerted by the snow that provides the necessary centripetal force. This normal force will have both a vertical and horizontal component. Since the skier is not changing their vertical position, the vertical acceleration at point B will be equal to zero.

3. The vertical forces acting on the skier at point B consist of the weight (mg) and the vertical component of the normal force.

weight = mass × acceleration due to gravity

Given that the mass of the skier is 63.7 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:

weight = 63.7 kg × 9.8 m/s^2 = 624.26 N

4. Since the vertical acceleration at point B is zero, the vertical component of the normal force must balance the weight of the skier:

vertical normal force = weight = 624.26 N

Therefore, the normal force exerted by the snow on the skier at point B is 624.26 N.

To find the normal force exerted by the snow on the skier at point B, we can start by applying the principles of circular motion.

When the skier reaches point B, they are moving along a curved path due to the dip in the snow's surface. In this situation, the net force acting on the skier is the centripetal force directed towards the center of the circular path. The normal force exerted by the snow on the skier at point B provides this centripetal force.

Here's the step-by-step process to find the normal force:

1. Find the centripetal acceleration:
The centripetal acceleration is given by the equation: a = v^2 / r, where v is the speed of the skier at point A (7.87 m/s) and r is the radius of curvature (11.5 m).

Substituting the values, we get:
a = (7.87 m/s)^2 / 11.5 m

Calculate the value of a.

2. Calculate the net force:
The net force acting on the skier is given by the equation: F_net = m * a, where m is the mass of the skier (63.7 kg) and a is the centripetal acceleration.

Substituting the values, we get:
F_net = 63.7 kg * (value of a from step 1)

Calculate the value of F_net.

3. Determine the normal force:
At point B, the normal force exerted by the snow on the skier provides the centripetal force.
Therefore, the normal force is equal to the net force (F_net) calculated in step 2.

Calculate the value of the normal force.

Following these steps, you should be able to determine the normal force exerted by the snow on the skier at point B.