A 69 kg skier encounters a dip in the snow's surface that has a circular cross section with a radius of curvature of 12 m. If the skier's speed at point A as shown below is 8.4 m/s, what is the normal force exerted by the snow on the skier at point B? Ignore frictional forces.
To find the normal force exerted by the snow on the skier at point B, we can start by understanding the forces acting on the skier. At point B, the skier is following a curved path, and there are two forces acting on the skier: the gravitational force (weight) and the normal force.
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is exerted by the snow on the skier.
To solve this problem, we'll use the concept of centripetal force, which is the net force acting on an object moving in a circular path. It is directed towards the center of the circular path and is responsible for keeping the object moving in that path.
The centripetal force is given by:
Fc = m * v^2 / r
Where:
- Fc is the centripetal force
- m is the mass of the skier (69 kg)
- v is the speed of the skier at point B (8.4 m/s)
- r is the radius of curvature (12 m)
In this case, the gravitational force (weight) is directed vertically downwards and has a magnitude equal to the skier's weight.
Now, since the skier is not moving vertically, the normal force and weight have to balance each other. Therefore, the normal force will have the same magnitude as the weight, but in the opposite direction.
The weight of the skier can be calculated using the formula:
W = m * g
Where:
- W is the weight
- m is the mass of the skier (69 kg)
- g is the acceleration due to gravity (9.8 m/s^2)
Now, we can find the normal force by equating the weight and the normal force:
W = Fc = m * v^2 / r
Plugging in the given values:
W = (69 kg) * (8.4 m/s)^2 / 12 m
Calculating the result:
W = 408.9 N
So, the normal force exerted by the snow on the skier at point B is 408.9 N.