A 54.9-kg ice skater is moving at 4.09 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius 0.803 m around the pole.
(a) Determine the force exerted by the horizontal rope on her arms.
Answer in N
(b) Compare this force with her weight by finding the ratio of the force to her weight.
To solve this problem, we can analyze the forces acting on the ice skater as she moves in a circular path.
(a) The force exerted by the horizontal rope on her arms is the centripetal force that keeps her moving in a circle. We can use the centripetal force equation:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the skater (54.9 kg)
v is the velocity of the skater (4.09 m/s)
r is the radius of the circular path (0.803 m)
Plugging in the given values:
F = (54.9 kg * (4.09 m/s)^2) / 0.803 m
F = (54.9 kg * 16.7281 m^2/s^2) / 0.803 m
F = 920.64729 kg⋅m/s^2 / 0.803 m
F = 1144.8227 N
Therefore, the force exerted by the horizontal rope on her arms is approximately 1144.8227 newtons (N).
(b) To compare this force with her weight, we need to find her weight first. The weight of the skater can be calculated using the formula:
Weight = mass * gravity
Where:
Weight is the gravitational force acting on the skater
mass is the mass of the skater (54.9 kg)
gravity is the acceleration due to gravity (approximately 9.8 m/s^2)
Weight = 54.9 kg * 9.8 m/s^2
Weight = 538.02 N
To find the ratio of the force to her weight:
Ratio = Force / Weight
Ratio = 1144.8227 N / 538.02 N
Ratio ≈ 2.13
Therefore, the ratio of the force exerted by the horizontal rope to her weight is approximately 2.13.