M08. Period of a System in Simple Harmonic Motion

 Goal

The purpose of this exercise is to verify the relationship between period and mass for a mass oscillating in simulated simple harmonic motion on a vertical spring. In doing so, you'll practice experimental and analytical techniques that will serve you in later experiments.

 Prelab

 Introduction

This is a follow up to the investigations of simple harmonic motion in M07t as well as a lead in to a hands-on lab on the simple pendulum. Let's assume that the period of a mass on a spring oscillating vertically depends on the object's g, namely, T = Cf(m,k,g), where C is a dimensionless constant. In the dimensional analysis below, the units on the left must be units of period. The units in brackets on the right are those of mass, spring constant, and acceleration. The exponents a, b, and c are unknowns but can be determined with a bit of algebra.

[s]¹ = [kg]^a[kg/s²]^b[m/s²]^c

Grouping units of m, kg and s,

[s]¹ = [kg]^a+b[s]^-2(b+c)m^c

The units on both sides of the equality must be the same; therefore, we can write three independent equations in the variables a, b, and c.

1 = -2(b+c)

0 = a+b

0 = c

With c = 0, we have 1 = -2b. Then b = -½ and a = ½.  Substituting back in to the original formula:

[s]¹ = [kg]^½[kg/s²]^-½[m/s²]⁰

Thus, the function f(m,k,g) is (m/k)^½. From this, we learn that the period is proportional to the square root of the ratio of mass to spring constant with no dependence on g. It may seen strange that the period of a vertically-oscillating mass doesn't depend on gravity, but that is what dimensional analysis tells us. Later, we will use conservation of energy to prove that gravity has no effect. For now, note that we haven't proven the relationship based on any physical principles. We've simply shown that the relationship is consistent with dimensional analysis of the variables. Note also that this process doesn't determine the value of the dimensionless constant. In general,  T = C(m/k)^½. In this lab exercise, you'll determine the value of C. 

 Preliminary Methods & Measurements

You'll use this animation for data collection. When you play the animation, the spring will oscillate continuously. You'll be making measurements of equilibrium position throughout this exercise, so you'll need a method for doing that. Note that the equilibrium position is the position exactly midway between the extremes of position. Each determination of equilibrium position will require two position readings and a calculation. Each determination of period will require measuring the time for 10 consecutive cycles of the motion. Assume that masses are known to 0.001 kg. These are the measurement tools available:

• the position of pointer on the vertical scale. You can change the axis scaling under the Window tab. Click Apply after making a change.
• the time readout under Outputs
• the position vs. time graph. You can change the axis scaling in the Graph window. Click Apply after making a change.

In WebAssign M08PMM, you'll carry out and check several measurements to insure that your measurement methods are sufficiently precise and accurate for the next part of the lab. Here's an overview of what you'll do.

1. Determine equilibrium position to a precision of 0.001 m and check your result.
2. Determine period to a precision of 0.001 s and check your result.
3. Determine the spring constant to 3 significant figures.

 Data

Recall that the goal of the lab is to determine the relationship between Period and __________ Mass . What will go in the blank, mass added to the platform or total mass including the platform? The platform itself has a mass of 0.050 kg. The key to making the correct decision is to answer for yourself whether the period is determined by the added mass or by the total mass. Make it clear in your variable labels in Logger Pro what decision you made.

Use the animation to determine the period of oscillation for a number of values of _________ Mass. At least 5 data points will be needed for a good fit. Enter your data in Logger Pro.

 Analyzing the Data

1. Plot a graph of Period vs. ________ Mass.

2. Examine your graph and decide how to re-express one of the variables in order to linearize the graph. Create a calculated column in Logger Pro for the values of the re-expressed variable.

3. Plot a second graph using the re-expressed variable. Assuming you made a good choice, fit your data with a linear regression. In a text box, prepare the matching table and write the equation of the fit.

4. In your text box, show how you use the equation of the fit to determine the value of the dimensionless constant, C.

5. What value of C is expected from theory? Check your textbook if you're not sure. Then calculate the experimental error between the value of C from step 4 and the value expected from theory.

 Submitting your Work

Check that your Logger Pro file is formatted appropriately. Then submit it according to the schedule

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