Tag Archives: circular motion

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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Circular Motion

We reviewed a another Force problem, force applied to an object from an angle. This force can be though of to have two components. We used Newton’s second Law, and broke it down into three equations. One for each axis (x and y) and one for The vector sum. We thought about how both x and y must sum to 0, for y there is only one force to be concerned about (Fexternal sin θ). But for the x direction we need to consider friction as well. Since it is pushing against our origin we think of Friction as a negative force. So Friction and the x force must sum to zero as well.

We needed one more thing to solve the problem; the magnitude of the friction force. To find it we multiplied the Normal force by the Coefficient friction of kinetic force. (mu μ). We rearranged our equations, and then divided by each other to combine them into one. And now we know how to calculate the Normal Force and the external Force. Since we know those two we can calculate other things we may need too.

Circular Motion
We just started to touch on circular Motion. If you swing an object in a circle, the magnitude of velocity is constant, but the direction changes all the time. At one time it is in one spot, then a short time later it is a little further along.

The magnitude of the radius at time 1 is equal to the radius at time 2. But the directional change is caused by a little vector called “Delta r” which pushes the direction of the vector along.

The change in time is expressed by “Delta t” And the velocity is the change in r (Delta r) over the change in time (Delta t)

Velocity is the tangent formed by Change in r over change in t. This makes Velocity almost a right triangle, when the distance in the change in time is very small.

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