Lab 1: Effect of Mass on Inertial Balance

We start this semester trying to prove a relationship between mass and the period. No, we are not talking about physiological processes, but more like a horizontal pendulum. As mass is added to the system, the period, or the time it takes some material under tension to move back and forth changes. We have several interesting apparatuses to aid us in the experiment:

The thin black metal strips that is the main character of this experiment is apparently called an inertial balance. One side of it is clamped to the table using a C-clamp, named after its shape--it uses a clamp screw tightened by a turning lever, gripping the table by the sheer power of friction. On the other end of the inertial balance is a red tray, which is used here to hold weights. The thin strips can be pulled to the side, causing tension, and pendulum movement. The little piece of masking tape at the end allows us to take advantage of the photogate to measure the period. The photogate consists of a light emitter and sensor, and whenever the strip of tape passes through and interrupts the light, a signal is sent to a LabPro experimental interface, and consequently interpreted by a computer running the Logger Pro software.

The LabPro, the plastic brick on the left, allows a variety of sensors to be hooked up through a standardized interface, then the information outputted via USB. These devices will form a core for most subsequent experiments.

We use a Logger Pro template initialized to take input from each back and forth movement picked up by the photogate, and depict it in a graph as dots over a time axis, representing the period of the swing. The average period over a 15 second capture is recorded in seconds. Then we increment the mass loaded onto the inertial balance, taping copper cylinders of 100g onto the red tray. We repeat this experiment until the total mass is 800g. Then we use several "unknown" weights, which will come in handy when it comes to calculate the correlation between the mass and period. Naturally, the period increases as the mass increases.

 

Periods, and masses of unknown weights read by a scale:

1. Motion Sensor: 0.4020s, 191g
2. Wallet: 0.3513s, 101g
3. Eraser: 0.2979s, 18g

It is reasonable to presume that the mass-period relationship could be modeled by a power function. Thus, we presume that, where T is the period in seconds, and m is the mass in kilograms, and n is some power constant:

T = mⁿ

 It is also likely that the curve isn't ideal, not to mention that the units of measurement on both sides of this equation do not match. So we introduce a coefficient, A, such that:

T = A * mⁿ

And since the period is dependent on the mass of the entire system, which includes the mass of the inertial balance on top of our weights:

T = A * (m + M_tray)ⁿ

Now, since in Logger Pro, it looks as if we could derive a linear best-fit line, we could manipulate this equation into linear form so that we could more easily use that linear best-fit equation to eliminate some variables. One of the ways to do this is to take a natural logarithm of both sides, and use the laws of logs to split the right side into two terms:

ln T = n * ln (m + M_tray) + ln A

This looks very similar to the linear equation form y = mx + b. So when we use the software to derive a linear best-fit line, we could use n to be the slope, and ln A to be the y-intercept. That leaves the mass of the tray, which we will guess by noting that the correlation of the best-fit line remains the maximum 0.9999. We will be able to find a range of possible M_tray values to this significant figure.

But first, we use Logger Pro to create two more columns in our Data Table, using the expressions ln T and ln (m + M_tray) , which we use to construct a graph that will serve to be the basis of our best-fit line. M_tray is a constant that we change to see how the correlation is affected.

After some trial and error, we find that the range of values of M_tray, such that the correlation remains 0.9999, is 0.285kg-0.295kg. Using the best-fit equation and solving for A in ln A, we find that the ranges of n and A are 0.6555-0.6845 and 0.6523-6452, respectively.

• Maximum: T = 0.6452 * (m + 0.305)^0.6845
• Minimum: T = 0.6523 * (m + 0.285)^0.6555

And using a calculator (we used Wolfram Alpha) by plugging in these values in the equation T = A * (m + M_tray)ⁿ, and using the period measured from before, we are finally able to affirm the range of the unknown masses.

1. Motion Sensor:
   Maximum: 0.4020 = 0.6452 * (m + 0.305)^0.6845
   Minimum: 0.4020 = 0.6523 * (m + 0.285)^0.6555
2. Wallet:
   Maximum: 0.3513 = 0.6452 * (m + 0.305)^0.6845
   Minimum: 0.3513 = 0.6523 * (m + 0.285)^0.6555
3. Eraser:
   Maximum: 0.2979 = 0.6452 * (m + 0.305)^0.6845
   Minimum: 0.2979 = 0.6523 * (m + 0.285)^0.6555

As shown in the photo above, we have calculated for the maximum and minimum masses for each object. We add these up and divide by two to get the mean. The number in parentheses is the mass weighed from the balance scale. The difference between a maximum and mean value is the deviation.

1. Motion Sensor: 194.4g ± 1.6g (191g)
2. Wallet: 105.3g ± 1.2g (101g)
3. Eraser: 18.0g ± 0.4g (18g)