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Experiment 12 Simple Harmonic Motion

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User Generated
Subject
Physics
School
Central Florida Institute Palm Harbor
Type
Homework
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1/11
Experiment 12: Simple Harmonic Motion
Purpose:
The purpose of this lab is to explore the relationship between oscillations, masses, springs and
spring constants. In addition, this lab will also explore linear regression, interpretation of graphs
and percent errors.
Theory and Concepts:
When a force “F” is applied to a spring, the spring will extend a distance “x” according to
Hooke’s law which states the extension is directly proportional to the force applied to the
spring.
Hooke’s law is F = -k*x
When the force is generated by a weight hanging in a gravitational field, Hooke’s law becomes
F = m*g = -k*x
Rearranging and noting that “g” is an acceleration force which the second derivative of a
distance function


  
Solving this differential equation gives
x(t) = A*cos(ωt)
when x = 0, cos(ωt) = 1 so that x(0) = A which means the initial length of the spring = A.
Differentiating x(t) = A*cos(ωt) twice gives the following





  
So that
Taking the first derivative of x(t) gives




Rearranging and integrating from (t, x) = (0, A) to (T/4, 0) where T = 1 complete revolution
(we’re assuming our oscillation is ¼ of a complete rotation.

 



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  







  


  


And we know that cos(ωT/4) = 0 when ωT/4 = 90° = π/2 so that

Since ω
2
= k/m

And finally

 


 

So that
(1) A plot of mg vs x will have slope = -k and intercept = 0
(2) A plot of T^2 vs mass weights will have slope = 4pi^2/K and intercept 4pi^2/k * mass
spring.
Experimental Apparatus:
The apparatus for part 1 of this experiment consisted of a mass, meterstick and a spring as in
the following image
In part 2 of this experiment, the spring was allowed to oscillate up and down and the time for an
oscillation was measured as in the following image.

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Experiment 12: Simple Harmonic Motion Purpose: The purpose of this lab is to explore the relationship between oscillations, masses, springs and spring constants. In addition, this lab will also explore linear regression, interpretation of graphs and percent errors. Theory and Concepts: When a force “F” is applied to a spring, the spring will extend a distance “x” according to Hooke’s law which states the extension is directly proportional to the force applied to the spring. Hooke’s law is F = -k*x When the force is generated by a weight hanging in a gravitational field, Hooke’s law becomes F = m*g = -k*x Rearranging and noting that “g” is an acceleration force which the second derivative of a distance function 𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑑2 𝑥 𝑘 = (− ) ∗ 𝑥 2 𝑑𝑡 𝑚 Solving this differential equation gives x(t) = A*cos(ωt) when x = 0, cos(ωt) = 1 so that x(0) = A which means the initial length of the spring = A. Differentiating x(t) = A*cos(ωt) twice gives the following 𝑑2 𝑥 𝑘 = −𝐴𝜔2 cos(𝜔𝑡) = −𝜔2 𝑥(𝑡) = (− ) ∗ 𝑥 2 𝑑𝑡 𝑚 So that 𝑘 𝜔2 = ( ) 𝑚 Taking the first derivative of x(t) gives 𝑣(𝑡) = 𝑑𝑥 = 𝐴𝜔 sin(𝜔𝑡) 𝑑𝑡 Rearranging and integrating from (t, x) = (0, A) to (T/4, 0) where T = 1 complete revolution (we’re assuming our oscillation is ¼ of a complete rotation. 0 𝑇 4 ∫ 𝑑𝑥 = 𝐴𝜔 ∫ sin(𝜔? ...
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