A 55.0 kg ice skater is moving at 4.04 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 0.795 m around the pole.
(a) Determine the force exerted by the horizontal rope on her arms.
(b) Compare this force with her weight by finding the ratio of the force to her weight.
Why did the ice skater grab the rope tied to a pole? Because she thought she could "pole" vault her way to victory! Let's find out the force exerted by the rope on her arms and compare it to her weight.
(a) To determine the force exerted by the rope, we can use the centripetal force formula:
F = m * (v^2 / r)
where
F is the force,
m is the mass (55.0 kg),
v is the velocity (4.04 m/s), and
r is the radius (0.795 m).
Substituting the values into the formula, we get:
F = (55.0 kg) * (4.04 m/s)^2 / 0.795 m
Calculating this, we find:
F ≈ 552.28 N
Therefore, the force exerted by the horizontal rope on her arms is approximately 552.28 Newtons.
(b) To compare this force with her weight, we need to find her weight first. Her weight can be calculated using the formula:
Weight = mass * gravity
where
mass is the mass (55.0 kg), and
gravity is approximately 9.8 m/s^2.
Substituting the values into the formula, we have:
Weight = (55.0 kg) * (9.8 m/s^2)
Calculating this, we get:
Weight ≈ 539 N
Now, let's find the ratio of the force to her weight:
Ratio = Force / Weight
Substituting the values we found earlier, we get:
Ratio ≈ 552.28 N / 539 N
Calculating this, we find:
Ratio ≈ 1.026
So, the ratio of the force exerted by the horizontal rope on her arms to her weight is approximately 1.026.
Isn't it amazing how physics can make us "skate" around numbers and calculations? Now go out there and "pole"ish your own knowledge!
To solve this problem, we can use Newton's Second Law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
(a) To find the force exerted by the horizontal rope on her arms, we need to figure out the acceleration of the ice skater. Since she is moving in a circle, she is experiencing centripetal acceleration, which is given by the equation:
a = v² / r
where:
a = centripetal acceleration
v = velocity of the ice skater
r = radius of the circular path
Given:
m = 55.0 kg (mass of the ice skater)
v = 4.04 m/s (velocity of the ice skater)
r = 0.795 m (radius)
Substituting the given values into the equation, we can find the centripetal acceleration:
a = (4.04 m/s)² / 0.795 m
= 16.3216 m²/s² / 0.795 m
= 20.532 s²
Now, we can use Newton's Second Law to find the force exerted by the rope. The net force is equal to the centripetal force:
F = m * a
Substituting the known values, we get:
F = 55.0 kg * 20.532 s²
= 1129.76 N
Therefore, the force exerted by the horizontal rope on her arms is 1129.76 N.
(b) To find the ratio of the force to her weight, we need to divide the force exerted by the rope by her weight:
Ratio = F / Weight
Given:
Weight = mass * gravity
m = 55.0 kg (mass of the ice skater)
g = 9.8 m/s² (acceleration due to gravity)
Weight = 55.0 kg * 9.8 m/s²
= 539.0 N
Ratio = 1129.76 N / 539.0 N
= 2.098
Therefore, the ratio of the force exerted by the horizontal rope to her weight is approximately 2.098.
To answer these questions, we need to consider the forces acting on the ice skater.
(a) The force exerted by the horizontal rope on her arms can be found using Newton's second law, which states that the net force acting on an object is equal to its mass times its acceleration. In this case, the net force is provided by the tension in the rope. The centripetal force required to keep the skater moving in a circle can be provided by this tension.
The acceleration of the skater can be found using the centripetal acceleration formula:
a = v^2 / r
Where:
a = centripetal acceleration
v = velocity
r = radius of the circular path
Substituting the given values:
a = (4.04 m/s)^2 / 0.795 m = 20.67 m/s^2
The net force can be found using Newton's second law:
F_net = m * a
Where:
F_net = net force
m = mass of the skater
Substituting the given values:
F_net = 55.0 kg * 20.67 m/s^2 = 1136.85 N
Therefore, the force exerted by the horizontal rope on her arms is approximately 1136.85 N.
(b) To compare this force with her weight, we need to find the weight of the skater. Weight is given by the formula:
Weight = mass * gravity
Where:
Weight = force of gravity acting on the skater's mass
mass = mass of the skater
gravity = acceleration due to gravity (approximately 9.8 m/s^2)
Substituting the given values:
Weight = 55.0 kg * 9.8 m/s^2 = 539 N
To find the ratio of the force to her weight, we divide the force exerted by the horizontal rope on her arms (1136.85 N) by her weight (539 N):
Ratio = Force / Weight = 1136.85 N / 539 N = 2.11
Therefore, the ratio of the force exerted by the horizontal rope on her arms to her weight is approximately 2.11.