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Simple Harmonic Motion and Pendulums
SP211: Physics I
Fall 2018
Name:
1 Introduction
When an object is oscillating, the displacement of that object varies sinusoidally with time. Simple Harmonic Motion describes this oscillatory motion where the displacement, velocity and acceleration are sinusoidal. The objects we are most interested in today are the physical pendulum, simple pendulum and a spring oscillator.
We can model this oscillatory system using a spring. If we have a spring on the horizontal (one-dimensional motion), and using Hooke’s Law (F = -kx) we can use the following expression:
−k ∗ x = m ∗ a (1)
In order to solve this expression, we will rewrite it.
−k m x(t) =
d^2 dt x(t) (2)
The only solution that can satisfy this equation is a sinusoidal function. For today, we will be using:
xmaxcos(ω ∗ t + φ) (3)
ω =
2 π T
k m
In these equations, we have xmax is the Amplitude, so the maximum position in the positive or negative direction. ω is our angular frequency, and the portion ω ∗ t + φ is the phase of our wave. The φ is a phase constant, which describes the position of the wave in a cycle at t = 0 s. If we think about the conservation of Energy with a spring in motion, and we ignore heat and external forces, we can state that:
Emechanical =
kx^2 +
mv^2 (5)
If we set the initial equation when x = xmax (so v = 0) and the final when x = 0 (so v = vmax), we get the following expression: 1 2
kx^2 max =
mv^2 max (6)
Since we know ω^2 = (^) mk , we can reduce this expression to:
vmax = ωxmax (7)
2 Procedure: Simple Harmonic Motion of a Spring
2.1 Find the Mass of Spring and Rubber Bob
We won’t need this information immediately, but start the lab by measuring the mass of the spring and the rubber bob.
Mass of Spring = kg
Mass of Rubber Bob = kg
2.2 Measure the Spring Constant
Similar to Lab 4, we will start by measuring the spring constant k. For this part of the lab, attach the spring to the hanger, and measure the length of the spring without any mass attached to it. This is your x 0 , and record it in the appropriate column in the chart below for all masses. Data Table Mass Weight (m*g) Equilibrium Legnth (x 0 )
Spring Length (x) x-x 0 (m)
Use the graph below to plot the Weight (y-axis) versus the x-x 0 (x-axis). Draw a line of best fit, and then calculate the slope by making a 90◦^ triangle. This slope is your Spring Constant.
k = N/m
3.2 Calculations for Simple Pendulum
Calculate the theoretical period of your pendulum.
Length of Pendulum = m
Mass = m
T = s
Calculate the percent difference between the theoretical and experimental value.
% Difference = %
4 Rigid Pendulum
A physical pendulum is expected to swing with a period such that:
T = 2π
I
mgh
where m is the mass of the pendulum, I is the rotational inertia of the pendulum (including the use of the parallel axis theorem as needed), and h is the distance from the pendulum’s center of mass to the axis of rotation. For this metal meter stick, rotating about a point 10 cm from the end of the stick) is Irot = 0.11 kg ∗ m^2. Measure the time it takes for 10 oscillations when the physical pendulum is rotating around the 10cm mark and calculate the expected period.
mass of pendulum = kg
Theoretical Period = s
Data Table Trial Time for 10 periods (seconds) Time for 1 period (seconds) 1
AVERAGE
% Difference = %
5 Problem Solving: The Simple Harmonic Motion of a Spring
A 2.3 kg mass oscillates back and forth from the end of a spring of spring constant 120 N/m. At t = 0, the position of the block is x = 0.13 m and its velocity is vx = -3.4 m/s.
- What is the Angular frequency of the block?
- What is the mechanical energy of this block-spring system?
- What is the amplitude of the oscillation?
- What is the maximum speed of the block and where is this experienced over the motion?
- What is the phase constant? (Choose a cosine function to describe the motion)
- What is the position of the block as a function of time?
- What is the maximum acceleration of the block and where is this experienced over the motion?
