A 45-kg skater rounds a 5.0-m radius turn at 6.3 m/s. (A.) What are the vertical and horizontal components of the force the ice exerts on her skate blades? (B.) At what angle can she lean without falling over?
To solve this problem, we can start by understanding the dynamics of the skater going around the turn.
(A.) To find the vertical and horizontal components of the force the ice exerts on the skate blades, we need to consider the forces acting on the skater. The main force responsible for keeping the skater moving in a circular path is the frictional force between the skate blades and the ice. This force can be broken down into two components: the vertical component and the horizontal component.
The vertical component of the force is responsible for supporting the skater's weight and preventing her from sinking into the ice. The horizontal component of the force is responsible for providing the centripetal acceleration required for circular motion.
To calculate the vertical component, we can use the equation:
Vertical component = weight of skater = mass of skater * acceleration due to gravity
Vertical component = 45 kg * 9.8 m/s^2 ≈ 441 N
To calculate the horizontal component, we can use the equation:
Horizontal component = centripetal force = mass of skater * (velocity^2 / radius)
Horizontal component = 45 kg * (6.3 m/s)^2 / 5.0 m ≈ 357.42 N
Therefore, the vertical component of the force the ice exerts on the skate blades is approximately 441 N, while the horizontal component is approximately 357.42 N.
(B.) To determine the angle at which the skater can lean without falling over, we need to consider the balance of forces. When the skater leans, the center of gravity shifts towards the inside of the turn. At a certain angle, the sideways force on the skater will be equal to the maximum frictional force provided by the ice, preventing the skater from falling over.
The maximum frictional force can be calculated using the equation:
Maximum frictional force = static friction coefficient * vertical component
To find the static friction coefficient, we need to know the nature of the surface between the skate blades and the ice. Assuming it's relatively smooth ice, we can approximate the static friction coefficient to be around 0.05.
Maximum frictional force = 0.05 * 441 N ≈ 22.05 N
For the skater not to fall over, the horizontal component of the force (357.42 N) must be less than or equal to the maximum frictional force (22.05 N). Therefore, we can use trigonometry to find the maximum angle at which the skater can lean without falling over.
Using the equation:
Horizontal component = maximum frictional force * sin(angle)
angle = arcsin(Horizontal component / maximum frictional force)
angle = arcsin(357.42 N / 22.05 N) ≈ 89.05 degrees
So, the skater can lean at an angle of approximately 89.05 degrees without falling over.