23-Feb-2015 Lab 1: Deriving a power law for an inertial pendulum.

Lab 1

The purpose of this lab is to find the relationship between mass and period for an inertial balance .

To begin this experiment, we need to make sure we get all the material and stuff before we start.

1. A C-clamp to fix the inertial balance which can give a nice simple pendulum.(A simple pendulum would be a mass at the end of a string.)

2. A photogate and motion sensor, so they can connect each other and record the period after balance passes through the photogate.

3. Using a program such as LabPro to show you the period and diagram on computer.

4. Preparing some weights from 100 to 800 grams for testing different masses.

Here is a photo that show you what it is like when you finish setting up all the equipment.

1. It's the time to get start. (Check the equipment first see if they work)

In order to make this reduce the error of this lab, we need to measure different masses from 0 to 800.

then we collect and record the period without any weight, then put the weights from 100 to 800 grams.

The key to get the period is to record every time the tray passes through the photogate and get the average of period.

2. The idea to finish this lab is to set up a function, and then we can use our data to find the relationship between mass and period.

We have seen a lot of formulas during learning math or physics. Those formulas basically start with a equation, and then put into data and figure out the unknown.

For this type of question, lets give a power-law type of equation such as T=A(m+M)^n Which m represent mass of weight, and M represent the tray.

We can see that there are three unknowns we need to know: A, M, n.

After we take logarithm of each side:(Why we take logarithm both side is to make function more like a simple function such as linear function.)

ln T = n ln (m+M) + ln A which looks like y = m*x +b (ln T is y, n is slope, ln(m+M) is x, and ln A is constant b).

We know that equation y = m*x +b gives a straight line and also it's measurable.

Therefore, we can graph all the data into a diagram and get a straight line as possible as we can.

(X is mass, Y is period)

This is the image that ln T = n ln (m+M) + ln A as y = m*x +b
Therefore, trayM = 0.3 kg, n =0.6427 ln A = -0.4524 -> A =0.6361

3. Now we try different mass of tray, so we can get the correction close to 1.

For this one. The total mass is 0.2 when there is nothing on tray, so the tray weight is Mtray = 0.2kg

For this one. The total mass is 0.2 when there is nothing on tray, so the tray weight is Mtray = 0.28kg

For this one. The total mass is 0.2 when there is nothing on tray, so the tray weight is Mtray = 0.32kg

After try different Mtray, we got different correlation.

The reason why we try different mass of tray is to get max and min correlation, then we can get the range of the mass so that we can test our data.

For the maximum of correlation is when M = 0.20kg, and correlation is 0.9985, n=0.5083, ln A = -0.4104, A= 0.6634.

No.1 equation: ln T = 0.5083* ln (m+M) - 0.4104

the minimum of correlation is when M is around 0.32kg, and correlation is 0.9953, n= 0.6682, ln A = -0.4626, A = 0.6296.

No.2 equation: ln T = 0.6682* ln (m+M) - 0.4626

Extension:

To test this equations correctness: I have two objects which are calculator and phone. I measure weight, and I ues the simple pendulum and get their period.
(If I plug the period such as the phone's period into the two function we got and calculate the mass and get m1 and m2, If the the phone's mass mphone is in between m1 and m2, we see our fuction is fine.)

(original data of calculator's and phone's weight and period)

 Now I am going to plug period into equation No.1 and No.2 to get a range of the objects' weight.

(The unit of m should be kg)

This is the calculator result, and the mass we got is 0.2693 kg < m < 0.288 kg.

The calculator real mass is 0.2786 g, so we can see the range is correct.

(The unit of m should be kg)

This is the phone result, and the mass we got is 0.096 kg < m < 0.105 kg.

The calculator real mass is 0.1033 g, so we can see the range is correct.


After what I did, we can see that our data is fine.

Conclusion:

The lab is to find the relationship between mass and period, so I find the equation by trying different weights and collect all the periods, and then plug into a power-law standard equation and solve the equation.