# Simple harmonic motion rope swing problem

• Ertosthnes
In summary, to swing on a rope across a river, Jane needs a minimum initial velocity of 8.55 m/s and a minimum velocity of -76.92 m/s to make it back, with the negative velocity indicating that the wind provides enough energy for her to swing without needing an initial velocity.
Ertosthnes
Jane wants to swing on a rope across a river. What minimum speed does she need to make it across, and once she's across, what minimum speed does she need to make it back?

Here's what's given:
mass = 47 kg
horizontal wind - call it F - (opposite to her swing) = 120 N
horizontal distance (D) = 50 m
rope length (L) = 40 m
theta = 50 degrees

(hopefully you can see the image)

Here's how I started:

D = Lsin(theta) + Lsin(phi)

Plug in the values, and phi = 28.9 degrees

Then, Change in height = Lcos(phi) - Lcos(theta)
Plug in the values, change in height = 9.3 m

From there:

PE(o) + KE(o) + wind = PE(f) + KE(f)

mgh(0) + (1/2)mv(0)^2 - F(w)*D = mgh(f) + (1/2)mv(f)^2

and v(0) = 8.55 m/s

I got that part right. But, now how do I find the minimum velocity to go back? I tried switching the h(0) and h(f) in that last equation to go back, but it didn't work. What's wrong?

Have you considered that she is now swinging with the wind?

Yes, I did. Here's what happens:

mgh(0) + (1/2)mv(0)^2 + F(w)*D = mgh(f) + (1/2)mv(f)^2

46(9.8)(-9.3) + (1/2)(47)v(o)^2 + 120(50) = 0

and v(o)^2 = -76.92

But the fact that it's negative makes me think that it's wrong. Can I still take the square root of it?

Edit: Apparently I can. I just tried it again and got it right.

Last edited:
But if you got a negative velocity, it means that the wind provides enough energy to Jane to reach that point, so that she doesn't need any initial velocity.

That's weird. Are you sure the negative doesn't just mean that Jane is swinging in the opposite direction? Is there something I should have done to make the velocity squared positive?

## 1. What is simple harmonic motion and how does it relate to a rope swing problem?

Simple harmonic motion refers to the repetitive, back-and-forth motion of an object around an equilibrium position. In the case of a rope swing problem, the rope acts as the object in motion, swinging back and forth around a fixed point.

## 2. What factors affect the frequency and period of a rope swing in simple harmonic motion?

The frequency and period of a rope swing in simple harmonic motion are affected by the length of the rope, the mass of the object on the end of the rope, and the force of gravity.

## 3. How is the amplitude determined in a simple harmonic motion rope swing problem?

The amplitude of a rope swing in simple harmonic motion is determined by the initial angle at which the swing is released. The greater the initial angle, the greater the amplitude of the swing.

## 4. What is the relationship between the frequency and period of a rope swing in simple harmonic motion?

The frequency and period of a rope swing in simple harmonic motion are inversely related. This means that as the frequency increases, the period decreases, and vice versa.

## 5. How can the energy of a rope swing in simple harmonic motion be calculated?

The total energy of a rope swing in simple harmonic motion can be calculated using the equation E = (1/2)kA^2, where k is the spring constant and A is the amplitude of the swing. The kinetic energy can also be calculated using the equation KE = (1/2)mv^2, where m is the mass of the object and v is the velocity of the swing at a given point.

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