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Simple pendulum theory (Simple Harmonic Motion), Lecture notes of Advanced Physics

Vanderbilt University (Vandy)Advanced Physics

Simple pendulum theory in describes their objectives, equipment, theory and procedures.

Typology: Lecture notes

2021/2022

Uploaded on 03/31/2022

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1
OBJECTIVE
a) Calculate the acceleration of gravity ‘g’ and compare with expected value by
analyzing the motion of a pendulum moving with Simple Harmonic
Motion(SHM).
b) Calculate the length of a pendulum so that it can be used a pendulum clock.
EQUIPMENT
1. 2-m length of string
2. 1 large support rod, 1 small support rod, and 1 clamp
3. hanger
4. stopwatch
5. 2-m stick
THEORY
Consider a pendulum of length ‘L’ and mass ‘m’. Suppose the pendulum is swinging and
at an instant in time its angular position is ‘θ’ with respect to the vertical. The Free-Body
diagram for the pendulum is shown below at this instant in time.
1. Show that by applying N2L in the tangential direction (Σ Ft = mat) and by assuming
small oscillations (small θ), the following equation must be satisfied:
2
20
dg
dt L
θθ

+=

 Simple Harmonic Equation
THE SIMPLE PENDULUM (Simple Harmonic Motion)
pf3

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OBJECTIVE

a) Calculate the acceleration of gravity ‘g’ and compare with expected value by analyzing the motion of a pendulum moving with Simple Harmonic Motion(SHM). b) Calculate the length of a pendulum so that it can be used a pendulum clock.

EQUIPMENT

  1. 2-m length of string
  2. 1 large support rod, 1 small support rod, and 1 clamp
  3. hanger
  4. stopwatch
  5. 2-m stick

THEORY

Consider a pendulum of length ‘L’ and mass ‘m’. Suppose the pendulum is swinging and at an instant in time its angular position is ‘θ’ with respect to the vertical. The Free-Body diagram for the pendulum is shown below at this instant in time.

  1. Show that by applying N2L in the tangential direction (Σ Ft = ma (^) t ) and by assuming small oscillations (small θ), the following equation must be satisfied:

2 2 0

d g dt L

Simple Harmonic Equation

THE SIMPLE PENDULUM (Simple Harmonic Motion)

  1. Confirm that the solution to this equation is given by:

θ ( ) t = θ m cos(ω t + φ)Solution to SHM Equation

Where,

θ(t) = amplitude of oscillation (rad) θm = maximum amplitude of oscillations from equilibrium (rad) g L

ω = (angular frequency in units of rad/s) It is a measure of how fast

the oscillations occur. t = time (s) φ = phase angle (determined by initial conditions) (rad)

  1. The cosine and sin function repeat every period T. Thus:

2 2

cos( ) cos[ ( ) )] cos( ) cos[( ) )] The sine and cosine repeat when their phase changes by 2. Thus, 2 2 2 2

m m m m

t t T t t T t t T

T

L

T

g g L

T L g

θ θ θ ω φ θ ω φ θ ω φ θ ω φ ω π ω π π π π ω

π

  1. The graph of T^2 vs. 4π^2 L will give a straight line with the slope related to the acceleration of gravity ‘g’.

PROCEDURE

  1. Setup apparatus
  2. Measure length of pendulum (for corresponding length) from pivot point to the center of mass of hanger.
  3. Measure the time for 10 oscillations and calculate the period. Repeat for same length for a total of 3 runs.
  4. Calculate the average period for the 3 runs.
  5. Repeat steps (2) – (4) for the length measurements indicated on the table below and fill in the data.

6. Make a graph of Tave^2 vs. 4 π 2 L using EXCEL and obtain the equation of the best

curve-fit.

  1. Calculate the acceleration of gravity from equation.