On Earth all objects accelerate towards the center of the planet with a constant acceleration due to gravity. We call this g and it's generally given a value of 9.81 m/s². We'll use this information to calibrate a force sensor, and then use the force sensor to estimate the unknown mass of another object.

Attach a metal rod to your ring stand. Then, attach the force sensor to the rod so that it is hanging down. Make sure it's high enough that your largest mass doesn't touch the lab table when hanging from the sensor.

The type of force sensor we're using really measures how far a force displaces a rod. A stronger force will move the rod more, while a weaker force will move it less. So, since the force sensor really gives us a distance, we need to measure a known force and use that to create a tool we can use to measure other forces.

You may have already seen a unit of measurement called a Newton (N). A force of 1 N that's applied to a 1 kg mass, will accelerate it 1 m/s for each second the force is applied. Another way to think of it is: if you had a mass of 1 kg and you pushed it so that after 1 second it was moving 1 m/s, and after 2 seconds it was moving at 2 m/s, you would have applied 1 N of force to that mass. So, 1 N = 1 kg m/s².

This is important to us because g is an acceleration of 9.81 m/s². So, if you hold a 1 kg mass in your hand, gravity is trying to accelerate it 9.81 m/s² towards the earth. If 1 N = 1 kg * 1 m/s² then 1kg * 9.81 m/s² = 9.81 N! That means that gravity pulls a 1 kg mass with a force of 9.81 N.
So it seems that we're saying that if we know the mass of an object in kilograms we can calculate how many newtons gravity will pull on it with. Let's test that out.

 

 

What about a different mass, like 500 g? Let's use what we just found out again. Remember that 500 g = 0.5 kg.

We're going to use the force sensor next. Please plug in your Arduino if you haven't already and attach a force sensor to analog pin 2.

Remember that the force sensor actually senses displacement, or how far a force has pushed it from its resting place. To translate its displacement reading into a force in Newtons, we need to build a model. To build our model, we'll sample some readings from the force sensor, plot those, and fit a line to them. That way when we take a new reading, we can use the equation of the line we've built to translate the displacement value to a force in Newtons!

First, we need to zero out the displacement value. When you use a scale to find the mass of something in a container, you put the container on first, zero it out, and then add the material you want the mass of. The same idea applies here. We want to treat the displacement of the force sensor as 0 when there's no mass hanging from it. To do that, just click Zero Displacement. Next, we'll take readings from several masses. We recommend using combinations that total 0.5 kg, 1 kg, 1.5 kg, and 2 kg. Here's the procedure we'll follow:

Procedure:

1. With no mass on the the sensor, zero the displacement.
2. Hang a combination of masses from the sensor.
3. Press Edit for one of the rows in the table.
4. Enter the mass of that you've attached to the sensor in the Mass box of the row you're editing.
5. Press Save. The current displacement and the value of Newtons will be filled in for you using our previous formula.

Don't worry about mistakes, you can edit a row as many times as you want. Every time you save a row, a point will be added to the plot to the right of the table.

Let's take stock of what we've done so far. First we came up with a general solution for getting a force in Newtons that a hanging mass applies due to gravity. Then we linked mass with displacement and used our general solution to link displacement with force by gathering data. That's great! We can now take one of the masses we used, hang it from the sensor, and get the force in Newtons. But what if we want to use it on different masses? Then we need a model that represents everything we've found so far. Our model should give us a force in Newtons when we provide a displacement from the force sensor.

One thing to look for in our data is that the points should arrange roughly in a line. That's because our equation for Newtons given mass is a linear equation. That's a fancy way to say that when you graph it, it makes a line. Any guess why? Well, for now, it's enough to know it's a linear equation, and that it should look like a line. That makes building a model pretty easy! You've probably already seen slope-intercept form for a line as: y = mx + b. We can use that to build a model. In our situation, y is force in Newtons and x is displacement from the sensor. That leaves us with m (slope, or the steepness of the line) and b (y intercept, or where the line crosses the y-axis). We can adjust these parameters to put a line anywhere we want. If we arrange it so that the line crosses, or is very close to, our data points, it will be a pretty good model. The closer it is overall, the better.

Some of our tools look the same as in the last section. But now we've added sliders to adjust the slope and intercept. You can move the handles of the sliders for large changes. For fine adjustment, you can click the handles and then press the left and right arrows on your keyboard. Try get the line to cross, or come very close to, all the points.

After the line is adjusted, zero out the displacement from the sensor again and hang a few masses from it. Try a combination you have a data point for. As you do, look at Force According to Model. This is the force in Newtons given by the model we've constructed. Does the value exactly match your data point? Why? Also, watch for the large point that slides along the line. This shows where the current reading from the sensor sits on the model. Now try a combination of masses you don't have a data point for. Does the value seem reasonable? If you didn't know the mass on the sensor, you could estimate it using your model. Have a classmate hang an unknown mass combination from the sensor and try to estimate it's total mass.

Some other things to think about: If our points don't lie on a line, but we know the equation is linear, what kinds of things could account for that? Can you think of any ways to find the closest possible line for a set of data points?