Lab 1: Mass vs Period for an Inertial Balance 2/23/15

PHYSICS LAB #1: Started on 2/23/15
Mass vs. Period
Annemarie Branks
Professor Wolf

Objective: This lab is intended to show that, for an oscillating system, the period is directly proportional to the mass. This means that as the mass increases so will the period due to the object's inertia. We will write an equation representing a mass and a period's directly proportional relationship. We can then use this equation to find unknown masses.

Set up:

1. As shown in the photo below, use a C-clamp to secure the inertial balance to the table. This should allow the inertial balance to oscillate horizontally.
   
2. Set up the photogate so the inertial balance oscillates through it. A long, thin piece of tape should be placed on the end of the inertial balance since the balance may be too large. The photogate emits light and will able to detect when that light is obstructed. Once the photogate is hooked up to a computer, it will tell us how often that light is being obstructed by the tape when we make the inertial balance oscillate.
   

Procedure:

1. With no extra mass on the tray, pull back the inertial balance. Take notice of how much force you are using to pull back the balance, for you must use the same force for each trial in order to obtain accurate results. Record the period of the balance.
2. For each additional trial, add an extra 100 grams to the balance and record the period. In this experiment we repeated the trail until 800 grams. Multiple trials will allow us to create a more accurate curve. Always use the same amount of force when pulling back the inertial balance.

The image above shows the 100 gram discs taped to the inertial balance. The tape that is keeping the discs together may affect your results but not enough to be significant.

     3.  Input your data to create a Mass vs. Period graph. Notice that the zero mass, like in the graph below, has a period. Theoretically, if an oscillating system has no mass then there is no period, but obviously after letting go of the balance, that your system does oscillate and the photogate is picking up a period. This is due to the inertial balance having a mass of its own which will need to be accounted for when writing an equation.

     4.  Since we do not know the mass of the inertial balance's tray, we are going to use the User Parameter function on LoggerPro to approximate what it may be. The goal is to find masses that have a correlation as close to 1.000 as possible so we can achieve a straight line. Multiple values for the mass of the tray may have that correlation so we are going to record the largest and the smallest mass with a correlation that is the closest to 1.000. In our experiment, we ranged the mass of the tray to be from 0.280 kg to 0.360 kg with a correlation of 0.9996.
     5.  A typical linear equation looks as such: y = mx+b

     We know our y-values will be our period (T), our slope will be some constant (A), and the x-values with be the additional masses including the mass of the tray "(m+Mtray) that need to be set to a power (n) as indicated by the curvature of the line. The "b" will become zero since like we have started before, a zero mass will have zero period.
     Now we are able to set up an equation that is related to our data: T = A(m+Mtray)^n

     6. To figure out what the values of A and n are we must log both sides of the equation above. This is our new equation: lnT = nln(m+Mtray) + lnA
     7.  Change the graph from Period vs Mass to ln T vs ln (m+Mtray) so that we may find the values that correspond with our new equation. The n will be our new slope, ln(m+Mtray) will be our new x, and lnA will be our new y-intercept.

     8. Adjust the parameter Mtray to achieve a straight line. The goal is to find a range of values that give a correlation coefficient of 0.9999. Record the maximum and minimum values of Mtray that produce a correlation as close to 1.000 as possible. For my group we found a minimum mass of 0.280 kg and a maximum mass of 0.360 kg.

     9. Record the slope and y-intersection for the maximum and minimum Mtray values. For our experiment, specifically, we found that a minimum mass of 0.280 kg resulted in a slope of 0.6465 and a y-intercept of -0.4336 seconds. The maximum mass of 0.360 kg resulted in a slope of 0.7424 and a y-intercept of -0.4852 seconds. The range from 0.280 kg and 0.360 kg had a correlation of 0.9996.

  10. With our new information we can complete the equation that will describe the relationship between mass and period; however, we will have two equations since we have a maximum and minimum mass for the tray of the balance.By plugging in we first has the equations

• minimum: lnT = 0.6465ln(m+Mtray)-0.4336
• maximum: lnT = 0.7424ln(m+Mtray)-0.4852

One we wrote those equations we were able to e both sides and write the equations in terms of T. They appear as follows:

• minimum: T = 0.648(m+0.280)^0.6465
• maximum: T = 0.616(m+0.360)^0.7424

     11. To see how well the new equations describe the mass and period relationship we are going to do tests of unknown masses.We will then compare their theoretical masses with their actual masses. We repeated the experiment using a white box and a water bottle. The white box had a period of 0.3919 seconds and the water bottle had a period of 0.6652 seconds. With our new equations we found that the white box has an approximate range for its mass which is 0.179 kg - 0.184 kg. The water bottle had a range of 0.761 kg - 0.749 kg.

     12.  Weigh the objects on a scale for find their actual masses and determine how close they are to their theoretical ranges. Determine if your equation accurately determines the mass of objects.

Specifically, for our experiment, the white box is 0.172 kg which is close to the minimum value but not within range. The water bottle is 0.601 which seems to be significantly under the minimum value of its predicted range.

Summary:

     Finding the periods of eight known masses and the unknown mass of the tray allowed us to plot points on a graph and create two equations that account for the lowest and highest possible values for the mass of the tray. These equations are meant to describe the relationship between period and mass. To determine the accuracy of our equations we ran two more experiments of unknown masses. We then compared the objects' theoretical masses and actual masses. It was found that though this method gave close results, the results are not very accurate, yet the experiments did visually prove that there is a relationship between mass and period.

Error: 

• Each trial required a person to pull back the inertial balance with a certain amount of force every time. If someone had not pulled the inertial balance enough, then the period would have been too small to fit in a trend amongst the other periods. I am sure my lab partner did his best to pull back the inertial balance with the same force for every trial, but it is most likely he did not use the same exact force every time. This could have affected our results.
• Collecting data for the period of the water bottle, we laid the water bottle down length wise. This allowed more room for the water to be moving back and forth, but the water would not have been moving in sync with the inertial balance. The force of the water in the bottle would have affected the period thus affecting our predicted value for its mass.