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Theory
Theory
What causes the motion of a mass-spring oscillator?
What causes the motion of a mass-spring oscillator?
The motion of the mass is caused by the force exerted on it by the spring.
In the initial equilibrium position, the force exerted on spring by the mass is due to gravity (mg). The force of gravity (or the weight) is balanced by the force exerted on it by the spring.
Suspended mass stretches the spring, which is not a natural state for the spring. So, the reaction force of spring is pulling the suspended mass up.
The equation below describes an ongoing struggle between the two forces, weight and the spring's reaction:
- ky = mg
An interesting situation occurs when the spring is compressed. Due to inertia, the mass pulled up by the spring does not stop immediately at its equilibrium position but travels farther and actually compresses the spring. Now, the spring "does not want" (I phrased it as the spring could want something) to be compressed, so it acts against the inertia of the compressing mass. Do you see the meaning of minus? Always against (opposite)!
Figure 1.
The properties of the spring (constant k) and the mass (m) cause the oscillating motion. Now we will describe the motion with the period of oscillation in the following equation:
According to the formula above, the frequency of oscillation grows as the stiffness of the spring (1/k) increases. Notice that reducing the mass attached to the spring will have the same effect.
The first equation
- ky = mg
is called the oscillator formula. It is a very interesting equation from the mathematical viewpoint: it compares position (y) with its second derivative, acceleration (g).
- ky = m (d²y/dt²)
or
d²y/dt² = - k/m (y(t))
The natural solution to an equation like that is the sin function:
y = A sin ωt
where ω² = k/m
Laboratory Procedure
Laboratory Procedure
Hook one of three springs on the force sensor as shown in the picture. Connect the sensor plug-in to the LabQuest interface. Suspend a 10 g weight on a spring.
Figure 2.
Gently pull the weight about an inch (or ab. 2 cm) down and release it. Go to the LabQuest graphing mode and start recording the graph.
Repeat the same for several other weights. Make sure that the weights do not deform the spring.
You should get a graph similar to the one shown in the picture (Figure 4). Write down the coefficients.
Figure 3. LabQuest screenshot of an oscillation
To find the coefficients of an oscillation, analyze the best fit graph. Follow the sequence:
Analyze -> Curve Fit -> Choose Fit -> Sine -> Coefficients
as shown in the pictures.
Figure 4a. Reading the coefficients
Figure 4b. Reading the coefficients
Data Analysis
Data Analysis
Based on the data tables
- Is there any pattern that indicates relationships between columns, rows, or tables?
- Is there any relationship between those coefficients and suspended masses?
- Is there any relationship between those coefficients and the length of springs?
- How would you interpret the coefficients?
Precalc review: screenshot of a note on the trigonometric function coefficients (Sullivan, E (2018) Precalculus, 10e, p. xxx)
Results
Results
Find the period of oscillation (T) based on the established value of k.
Find the period of oscillation (T) based on the proper coefficient.
Compare the results.
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