How Much Does the Rope Stretch When a Circus Performer Hangs at Rest?

[joeypeter]

A 55.0 kg circus performer oscillates up and down at the end of a long elastic rope at a rate of once every 2.60 s. The elastic rope obeys Hooke's Law. By how much is the rope extended beyond its unloaded length when the performer hangs at rest?

Can someone help me to find the answer step by step?

 

[mike115]

The period of the system (since it obeys Hooke's law) is given by T = 2*pi*sqrt(m/k). You can use the given data to solve for k, the effective spring constant.

When the performer is at rest, the net force on him must be zero. Therefore, the upward spring force, kx, must equal the gravitational force on the person.

 

[Moetasim]

Sure, I can help you find the answer step by step.

Step 1: Identify the given information.
- Mass of the circus performer = 55.0 kg
- Rate of oscillation = 1 cycle every 2.60 s
- The elastic rope obeys Hooke's Law

Step 2: Convert the given rate of oscillation into frequency.
- Frequency = 1 / time period = 1 / 2.60 s = 0.3846 Hz

Step 3: Use the formula for frequency of simple harmonic motion to calculate the angular frequency (ω).
- Frequency (f) = ω / 2π
- ω = 2πf = 2π(0.3846 Hz) = 2.423 rad/s

Step 4: Use Hooke's Law to calculate the spring constant (k) of the elastic rope.
- Hooke's Law: F = -kx
- Where F is the force applied, k is the spring constant, and x is the displacement from the equilibrium position.
- We know that the weight of the performer (mg) is equal to the force applied (F) when the performer is at rest.
- Therefore, mg = kx
- Solving for k, we get k = mg / x

Step 5: Calculate the displacement (x) of the rope when the performer is at rest.
- When the performer is at rest, the rope is at its equilibrium position, meaning there is no displacement.
- Therefore, x = 0

Step 6: Substitute the values into the equation for k.
- k = (55.0 kg)(9.8 m/s²) / 0 = 539 N/m

Step 7: Use the formula for displacement in simple harmonic motion to calculate the amplitude (A) of the oscillation.
- Displacement (x) = A sin(ωt)
- Where A is the amplitude and t is the time.
- Since we are looking for the displacement when the performer is at rest, we can set t = 0.
- Therefore, x = A sin(ω(0)) = A(0) = 0
- This means that the amplitude is equal to the displacement at rest, which is 0.

Step 8: Calculate the extension of the rope when the performer hangs at rest.
- The extension of the rope is the difference between the