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Velocity, Center of Mass and CIRCULAR MOTION

Velocity and Center of Mass

What happens if a rocket launches up into the air, then explodes? Where is the center of Mass? What about the velocities? Well, The velocity of the center of mass remains the same. But for instance, if you launch a rocket with two stages, and at the peak of the parabola, jettison the spent fuel canister? It’s velocity becomes zero, as the second stage kicks in and pushes off from the first stage. It picks up double the velocity (if they are equal weight, and is able to fly twice as far as the two masses would have gone together. The first stage falls straight down.


To keep track of Circular motion, we need to watch the same things we watch for Linear motion.

  • Position r (letter r)
  • Velocity ω (little omega)
  • Acceleration α (little alpha)

But now we are moving into Polar Coordinates, where it is much easier to keep track of a position on a circle in terms of radius and theta θ (Angle from the origin). If you use Cartesian coordinates you have to write x and y as r cos θ and r sin θ. So it’s easier to use Cartesian. Theta is also now always expressed in radians, or “length of arc over radius

What we do with the Polar Coordinates:

Theta in Radians equals length of arc over radius
1 ) Average Angular Velocity (little omega) equals change in Theta over change in Time.
2 ) Instantaneous Angular Velocity(little omega) equals the derivative of theta with respect to the derivative of time.
3 ) The derivative of Theta with respect to time is equivalent to the derivative of length with 4) respect to time multiplied by one over the radius.
5 ) The Tangential velocity at a time is equal to the derivative of the length with respect to time.
6 ) The Average Angular Velocity (little omega) equals the tangential velocity over the radius.
7 ) The Average Angular Acceleration (little alpha)  is equal to the change in Velocity (omega) over the change in time.
8 ) The angular instantaneous acceleration (little alpha) is equal to the derivative of angular velocity with respect to time.

You can use the same formula’s used for Linear constant acceleration, just replace the Linear velocity and acceleration with the Circular velocity and acceleration.

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Center of Mass

Finding the center of mass is essentially finding the weighted average of the component masses. Of course this can get more complicated depending on the number of masses and the amount of dimensions (1, 2 or 3).


Here cm means “Center of Mass”. You can expand this calculation out to x y and z, just consider the position (distance) of your mass from the origin in relation to the plane you are looking at.

For example an equilateral triangle, made from three masses connected by three straws. If you balance This contraption on a piece of cardboard, The center of mass along the x axis is Length over two and the center of pass along the y axis is 1/3(sqrt(3)/2) times L. The center of mass does not have to be where mass is.

The next part of the course combines lateral and circular motion and the center of mass is important for this.

Suppose you have a distributed mass of length L. And it’s a meter stick (just for fun). It has even linear density along the meter sticks, suppose you look at a very small quantity of mass, a little dm x distance from the origin (The start of the meter stick). If you add up all of those little bits of mass, and use the ratio dm/dx = M/L, you can find the linear density.

So, what this says, is that the center of mass, at some distance from the origin is the integral of the small change in mass times the distance, divided by the total mass. Which in the end figures out to the center of mass, is at one half the length.

Now think of a triangle. An Isosceles Right Triangle. Now to find the center of mass, divide it up into many little slices. (little dx’s and dy’s). Once again we can use the handy ratio. The little mass divided by the little area is equal to the big mass over the total area. dm/da = M/A.

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