Pendula and Springs Lab .docx

Pendula and Springs Lab .docx - Pendula and Springs Lab...

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Pendula and Springs Lab
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Abstract: This experiment was primarily designed to understand Hooke’s law through simple harmonic motion using springs and Pendula. The relationship between the motion of pendula and the effect of physical characteristics which are mass, length, and diameter on the period of the pendula were determined. The acceleration due to gravitation constant was calculated using the length and the period of the pendula. Also, the spring constant was measured using three different springs using their extension as a function of force with a fixed mass. The results obtain from the experiments performed in the lab showed that the period of the pendula has no relationship with various mass although the size of the balls also known as diameter and the length have affected the period of the pendula. The experimental gravitational acceleration was 9.631m/s 2 and the obtained spring constant of spring 1 is 6.639, spring 2 constant is 6.671, and spring 3 constant is 3.165. Introduction: The purpose of this lab was to understand the simple harmonic motion using Hooke’s law through the motion of pendula in the first part of the lab and the spring motions with a mass on the second part of the experiment. Hooke’s law stated that an object goes through simple harmonic motion if the restoring force is proportion to its displacement which is F =− kx . Newton’s second law state that force is equals to mass times its acceleration, so ma =− kx . By understanding this, the spring constant can be determined using the extensions of the springs and the oscillations of the mass and the time it takes to complete a full cycle. Also, the period of the pendulum can be measured through its length and using the equation that defines the angle created by the pendulum as it travels away from the vertical position (origin) and the restoring force by gravity.
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Theory: Calculation for Springs: Hooke’s law of springs state that: F =− kx Since F = ma in Newton’s second law, then F spring =− kx = ma Since acceleration is change in velocity as a function of time and velocity is the change in displace as a function of time which is a = dv dt = d 2 x dt 2 , Then: kx = m d 2 x dt 2 Using the displacement as a function of time at equilibrium position equation, x ( t ) = A cos ( ωt ) , d 2 x d t 2 can be solved and d 2 x d t 2 =− 2 cos ( ωt ) Then, Hooke’s law become kAcos ( ωt ) =− mA ω 2 cos ( ωt ) which in turn equals to ω = k m Calculation for Pendulum: The restoring force of the pendulum is due to gravitational force, so F = mgsinθ For small angles, approximately sinθ ≈θ , then F = mgθ
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