23-Feb-1025: Inertia Balance

Inertia Balance

Purpose 

The purpose of this lab is to find a mathematical equation that describes the behavior of an Inertial Balance. In order to do so, you are going to record and compare a range of periodic measurements.

Materials

For this lab you will use masking tape, a balance, a stand, a C-clamp, LabPro with power adapter and a computer with LoggerPro software installed. Most important you will  need an Inertial Balance, which is a device with a holding tray and two metal straps acting as springs. The balance will allow you to measure the mass of diferent objects by compararing their periods of oscillation once the balance is set into motion.

Lastly you need a Photogate, a black "C" shape sensor which transmits a light beam from one end of the "C" to the other. 

Set up

Above is the complete set up. First you need to secure one end of the inertial balance to the tabletop using a c-clamp. Then you need to put a thin piece of masking tape on the other end of the interial balance. Next, place the photogate on a stand so that when the balance is oscillating, the tape completely passes through the beam of light of the photogate. Lastly, connect the photogate to the  LabPro and plug it in to your computer. Make sure the power adapter from the LabPro is connected to an outlet. 

Procedure

1. By pulling and releasing the inertial balance (sideways) you are going to record the period of oscillation of an empty tray for 3 seconds. Record using LoggerPro (click on record button) and  a stop watch. Record the value of the period displayed in LoggerPro and compare it to value obtained by using a stopwatch. Then you are going to add 100 grams to the tray and measure and record the period. Successively add 100 grams until you reach 800 grams measuring and recording the period each time you add 100 grams. Record all of your data in a table like the one below 

2. Open a blank Logger Pro document and set a parameter Mtray for your guess for what the effective mass of the tray is.create a table.Then create a table. This table should contain 5 colums: Mass (Kg), Time (s), Mass+Mtray, ln Time and ln (Mass+Mtray) .

3. Plot a ln Time vs ln (Mass+Mtray). Adjust the value of the parameter Mtray until your graph gives you a beautiful straight line (linear fit 0.9999)

4. Explore different values of Mtray around the one you chose to estimate what the uncertainty is in the value of Mtray (ie. +/- 10). Record the value you chose and the upper and lower values of your estimate (each one yielding a beautiful straight line).

5. Record the curve fit equation that goes with each of these values so that each value of Mtray has its own y-intercept and slope.

6. Plug the values into  T=A(Mass+Mtray)^n we and notice that the relation between mass and period depends on a constant A.

7. T=period A=constant n=y-intercept

8.Take the natural logarithm of each side of the equation T=A(Mass+Mtray)^n to obtain lnT= n ln(Mass+Mtray) + ln A

9. Use A=e^lnA to determine the value of A

Now that you have found a mathematical model for the behavior of your internal pendulum, use the equation to find the mass of other unknown objects. 

Period obtained by using the equation.

Period obtained after obtaining the mass of each object from a scale.