A 59.0-kg ice skater is moving at 4.05 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 0.880 m around the pole.
(a) Determine the magnitude of the force exerted by the horizontal rope on her arms.
kN
(b) Compare this force with her weight.
Frope
W
=
To determine the magnitude of the force exerted by the horizontal rope on the ice skater's arms, we can use the centripetal force formula:
Fc = (m * v^2) / r
Where:
Fc = centripetal force
m = mass of the ice skater
v = velocity of the ice skater
r = radius of the circular motion
For the given values:
m = 59.0 kg
v = 4.05 m/s
r = 0.880 m
We can substitute these values into the formula:
Fc = (59.0 kg * (4.05 m/s)^2) / 0.880 m
Calculating this gives us:
Fc ≈ 117.13 N
Therefore, the magnitude of the force exerted by the horizontal rope on the ice skater's arms is approximately 117.13 N.
To compare this force with the ice skater's weight, we can use the equation:
Frope / W
Where:
Frope = force exerted by the rope
W = weight of the ice skater
For the given values:
Frope = 117.13 N
W = m * g, where g is the acceleration due to gravity (approximated as 9.8 m/s^2)
m = 59.0 kg
Substituting these values into the equation:
Frope / W = 117.13 N / (59.0 kg * 9.8 m/s^2)
Calculating this gives us:
Frope / W ≈ 0.197
Therefore, the force exerted by the rope is approximately 0.197 times the weight of the ice skater.