Introducing three main concepts - Energy, Inertia Moment and Angular Velocity. We describe what is the moment of Inertia, how do we calculate it and how do we measure it?
- 08 Nov 2015
The Moment of Inertia gives you an understanding of how difficult it is to stop an object once it is moving/rotating or to start moving/rotating an object once it has stopped. You can give it a value and this value depends on the weight and the speed of the object.
Previous video tutorials:
Now let's get into the theory. We have a cylinder and this cylinder is rotating on an axle and is rotating in a certain direction. Because of the rotation this cylinder has energy and energy is marked with the letter E and it has kinetic energy. Now there are different types of cylinders and different ways to calculate the kinetic energy of this cylinder and they all depend on something called the inertia moment and the the speed of rotation which is called omega. We have these 3 values, the kinetic energy the inertia moment and the speed of rotation of this cylinder. The connection between these values is the following. E is equal to a half of the inertia moment multiplied by omega squared. Omega marks the speed of rotation of this cylinder. This is the kinetic energy of our cylinder. For the inertia moment, it depends on the mass and radius of this cylinder. The inertia moment is equal to a half of the mass of the cylinder multiplied by the radius of the cylinder squared. The radius of the cylinder is right here and the larger the cylinder, the more the inertia moment, the larger the mass, the more the inertia moment and actually the inertia moment shows how difficult it is once this cylinder is rotating to stop it from rotating or once this cylinder is stopped to start it rotating. That's actually the notion behind the inertia moment.
This here is moment of inertia.
This formula right here applies only for a solid cylinder. If you look at the LEGO wheel that we are currently using it's a cylinder but it little bit different because the mass of this cylinder is not equally distributed. Our LEGO wheel has a tire and a rim. If you measure the tire and we've done it it's about 25 grams and the rim is about 15 grams and the whole wheel is about 40 grams. So the mass is not equally distributed and because of that it get very complicated and we'll use a very simple formula, it won't be very accurate, but it will be quite accurate for the principle that we would like to show. When you have a cylinder that has most of the mass here between the 2 radii. We have an outer circle and for it we have r outer and for the inner circle we have r inner.
We have the inertia moment for the cylinder that has an outer circle and an inner circle that is equal to a half of the mass multiplied by the radius of the outer circle squared plus radius inner squared. That's the formula that we are going to use for finding the energy, the inertia moment and from there the energy of our LEGO wheel that is rotating. We have the 2 radii and the mass. Last but not least we must find the speed of rotation. The speed of rotation of this wheel and the angular velocity is marked with omega. We measured the angular velocity last time.
We measured last time that our motor was rotating with 860 degrees per second. That was the angular velocity of our motor when it was reaching the power of a 100%.
If we know that one circle has 360 degrees how many rotations per second are we doing. If we divide 860 by 360
will give us the number of rotations that our motor is doing for 1 second. This here is equal approximately to 2.39. We have our motor doing 2.39 rotations per second.
Because we are working in the system international units we must have all the units in the system so that we can get the energy. For the energy we would like to get it in joules, that means that we must have the mass in kg and the mass of our wheel is 0.04 kg, the radius then must be in meters, the larger radius is 0.07 m, that's the radius of the outer circle.
We need the radius of the inner circle is equal to 0.049 m, and we must also have the omega, we must have it in radians per second. What does that means? We have a number of rotations per second
and we multiply this by 2 pi, because n 1 circle we have 2 pi radians. Below the video we'll give you more links for resources that explain exactly the conversion between the rotations per second and the radians per second If we multiply 2.39 by 2 pi we'll get something like this. So we have 2.39 multiplied by 2*3.14 and the result is about 15 radians per second. Our wheel is doing 15 radians per second.
Omega is actually 15 radians per second. Because the video is getting too long in the next video we'll continue substituting these values in the formula and getting the result in joules.