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Simple Harmonic Motion
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The Law of Conversation of Energy
The energy conservation in simple harmonic motion is E = K (kinetic energy) + U (potential energy).
The elastic potential energy of the cord (U) is 1/2kx^2. This is equal to the total energy of the system prior to the jump.
Apply the Knowledge:
The Physics Behind Bungee Jumping
The bungee jumper experiences gravitational force until his or her cord has reached its full length (L in the picture). At this point, there exists a downward gravitational force, as well as an upward restoring force exerted on the spring-like cord.
This force is subject to Hooke's Law:
F = -k(x)
The highest bungee jump in the world is 321 metres high, on the Royal Gorge Bridge.
A UBC Physics 101 student that weighs 50.0kg is attached to an unstretched 30.0m long cord, ready to jump from the Royal Gorge Bridge. She jumps and comes to rest right before the water surface on the first descent.
Assume the cord is subject to Hooke's Law and oscillates in simple harmonic motion.
What is the period of the oscillations?
Answer:
The Basics:
Applications
Simple harmonic motion can be found in many real-life examples:
- pendulums
- sound waves
- springs
To find the displacement of the cord from equilibrium (x):
The unstretched cord is 30.0m and we know that the jumper reaches the entire length of the jump, 321m. Therefore,
x = 321 - 30 = 291m.
To find the spring constant (k):
At the bottom of the descent, the bungee jumper is essentially static. The upward force of the cord would equal the gravitational force. Therefore, we know that kx = mg, and can solve for
k = mg/x = (50)(9.81)/(291) = 1.69
The period of a mass on a spring is calculated by:
T = 2pi x sqrt(m/k)
= 2pi x sqrt(50/1.69)
= 34.2s
Bungee Jumping!
We can observe that many things in daily life oscillate when stretched from a position in equilibrium. In bungee jumping, the cord used is highly elastic and acts very much like a spring.
A bungee jumper (the weight) is attached to the free end of the cord and jumps from a height. The weight will bounce back up and down, oscillating until it regains its point of equilibrium.