How to Measure Mass of the Earth Using Arduino.

In early nineteenth century scientists used Newton's second law and Newton's law of universal gravitation to measure mass of the Earth. These equations are F = ma and F=(GmM)/(r²) respectively(m=mass of the object, a=acceleration, G=Gravitational Constant, M=mass of the Earth, r=Radius of the earth). If we substitute g i.e. acceleration due to gravity for the 'a' acceleration term and combine these two equation we get.

mg = GmM/r²

This equation can be solved for M i.e mass of the earth.

M = (gr²)/G.

We will assume that we know the value of G and r. We will find g i.e. acceleration due to gravity using a small experiment, involving arduino and a few sensors. Finally we will put all things together to find the value of M.

If equations seems a bit awkward , look at the attached image.

Let's get started with the experiment.

12 More Images

Here is the list of things we will need for experiment to measure acceleration due to gravity.

• Any arduino board, preferably uno/duomilanove (no specific reason just that I have used this board for this experiment)
• USB cable for connecting arduino to PC
• 560 Ohm resistors (4 in quantity), 100 Ohm resistors (2 in quantity)
• piezoelectric sensor (4 in quantity)
• One pair of IR Tx (IR LED) and Rx(Photodiode)
• One potentiometer
• One opamp (like UA741C)
• One breadoard
• Three 1.2 meter long wires
• small 5cmx5cm general purpose PCB for assembling IR circuitry
• Tape
• Cardboard sheet

In case of any doubt look at the attached Images.

This circuit will be used to time-stamp the starting time of free fall for an object used to measure 'g'. Here are the important points about the fritzing schematic image.

• IR LED is used in forward bias which acts as just normal LED in this application i.e. illuminating the IR receiver Photodiode.
• Photodiode is used in reverse bias.
• Output from the junction of cathode and resistor is connected to one of the input terminal of opamp.
• Other input terminal of opamp is connected to potentiometer that determines the threshold voltage.
• Output of opamp goes to the arduino board.

Look at the attached image in case of any doubt, or refer the video (time= 5m 16s) .

This circuit will be used to time-stamp the ending time of free fall for an object used to measure 'g'. Here are the important points about the fritzing schematic image.

• 4 piezoelectric sensors are connected as shown in the schematic.
• 4 560 Ohm resistor are connected in parallel with the piezoelectric sensors to reduce the voltage spike generated by piezoelectric sensor when free falling object hits it's surface.
• Negative terminal of piezoelectric sensors is connected to circuit ground.
• Positive terminal of piezoelectric sensor is connected to analog input pins.

In case of doubt refer to attached image or video (time = 9m 2 sec).

Follow these instructions to put things together.

• Use card board to make roof like structure and stick it to the wall at around 1.1 meters.
• Attach IR module facing downward to cardboard roof.
• Connect output of opamp to pin no 4 of arduino using one of the 1.2m long wires.
• Connect Vcc and GND from arduino to IR module using rest of the 1.2m long wires.
• Make sure that things are firmly attached wires to the wall, IR module to the cardboard roof, piezoelectric module to the ground just below IR module.

In case of doubt, look at the attached image, it is properly marked for ease of understanding the assembly.

With all things at place (except arduino code, discussed in step 8), lets go through the procedure of this experiment.

• Stick a small thin projection on top of the object used for the free fall as shown in the image numbered as 1.
• Before we leave our object to carry on free fall, hold it just at the 1 meter mark, such that projection on the top obstruct the light from IR to photo diode. As shown in the image numbered as 2.
• leave the object without imparting any force to let it fall free on the piezoelectric landing zone.
• As it falls down there is no obstruction in the path of IR TX and RX, giving us the signal about just left object in term of starting time of free fall.
• As object touches the landing zone, piezoelectric sensors generate controlled voltage spike notifying the arrival of object in term of ending time of the free fall.
• If we subtract ending time from the starting time we get time of flight that we will use to calculate the acceleration due to gravity as shown in next step.

Here again we will use Newton's equation that goes like -

Distance traveled = (initial velocity * time of travel) + (acceleration * (time of travel)²)/2

In our case these variables takes following values-

• Distance traveled = 1 Meter
• initial velocity = 0 m/s as we are starting from rest
• acceleration = g

So equation reduces to

1 = (g * (time of travel)²)/2

So,

g = 2 / (time of travel)²

So after we do the experiment we will use this equation to calculate value of 'g' and find mass of the earth. In case of any doubt refer the attached image or video.

I am attaching the arduino code for this experiment. Code is really simple. In case of any doubt refer the video where I explain the code in detail (time in video = 13 minute 35 secs)

Following things should be taken care before doing this experiment.

• Threshold of the piezoelectric sensor should be properly set in the arduino code.
• Place piezoelectric panel just below the IR module.
• Make sure you drop the object from 1 Meter with out imparting any external force.

Here I am doing this experiment at time stamp 17 minute in the attached video. We will calculate Mass of earth using results i.e. time of flight obtained from this experiment.

After repeating this experiment for 3 times I got following results in terms of Time of Flight.

• 443 ms
• 443 ms
• 464 ms

Lets take average.

Average time of flight = 450 ms

If we substitute this value in equation to obtain acceleration due to gravity we get acceleration to be around 9.8755 m/s². As seen in the attached image.

Finally substituting this value of 'g' to get mass of earth we get it to be around 6.0066 x 10²⁴ Kg, as you can see in the attached image.

Actual mass of the Earth as per google is 5.972 x 10²⁴ Kg. I think we got close.

Thank you for your time, In case of any doubt refer the video or put it in the comments I will be happy to help.