You and a friend are going to a river, where they have a rope swing. Your friend, who has a mass of 60 kg, hops on the end of the rope. You tug horizontally on your friend with a force of 300 N, pulling them back until you can't move them any more. Then you let go and watch them swing out over the river. When the rope swing is at it's farthest point your friend lets go, eventually splashing into the river. If the rope is 30 m long, what total distance in m did your friend travel from her starting point to where she let go?
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To determine the total distance your friend traveled from her starting point to where she let go, we need to analyze the motion of the swinging rope.
When you tug horizontally on your friend, you are applying a force that acts in the opposite direction of the swing. This force creates tension in the rope, which acts as the centripetal force keeping your friend in circular motion.
First, we need to find the tension force in the rope when you tug on your friend. We can use the formula for centripetal force:
F = m * a
where F is the centripetal force, m is the mass of your friend, and a is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula:
a = (v^2) / r
where v is the velocity and r is the radius of the circular motion, which is equal to the length of the rope.
Rearranging the equation for a, we have:
v = sqrt(a * r)
Substituting the given values, we have:
v = sqrt((300 N) / (60 kg) * (30 m))
Next, we need to find the maximum height reached by your friend during the swing. At the farthest point, when your friend lets go, all of the gravitational potential energy will be converted into kinetic energy.
The gravitational potential energy (PE) can be calculated using the formula:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height.
Since the final kinetic energy (KE) is equal to the gravitational potential energy, we have:
KE = (1/2) * m * v^2
Setting the expressions for PE and KE equal to each other, we can solve for h:
m * g * h = (1/2) * m * v^2
h = (1/2) * v^2 / g
Substituting the calculated value for v, we have:
h = (1/2) * (sqrt((300 N) / (60 kg) * (30 m)))^2 / 9.8 m/s^2
Finally, to calculate the total distance traveled by your friend, we consider the circular path followed during the swing. The total distance traveled is equal to the circumference of the circle traced by swinging:
total distance = 2 * π * r
Substituting the given value for the length of the rope, we have:
total distance = 2 * π * (30 m)
Simplifying, we find the total distance traveled by your friend from her starting point to where she let go.