1. Uniform Circular Motion
Lecture 6
Uniform Circular Motion

2. Derivation
There is also a derivation slightly different, found in the text.
We currently understand circular motion by:
𝑥=𝑣×𝑡
when we have uniform motion.
In uniform circular motion, however, we use
𝜃=𝑣×𝑡
In angular motion we use
θ=𝜔×𝑡

3. When given a circle: Where R is given as the radius (r). We know that:
𝑥=𝑅𝑐𝑜𝑠𝜃=𝑅𝑐𝑜𝑠(𝜔𝑡)
𝑦=𝑅𝑠𝑖𝑛𝜃=𝑅𝑠𝑖𝑛(𝜔𝑡)
You can then take the derivative of the position in respect to time:
𝑑𝑥 𝑑𝑡 = 𝑉 𝑥 =−𝑅𝜔𝑠𝑖𝑛𝜔𝑡
and
𝑑𝑦 𝑑𝑡 = 𝑉 𝑦 =𝑅𝜔𝑐𝑜𝑠𝜔𝑡
You can then take the derivative a second time:
𝑑 2 𝑥 𝑑 𝑡 2 = 𝑎 𝑥 2 =−𝑅 𝜔 2 𝑐𝑜𝑠𝑡𝜔𝑡
and
𝑑 2 𝑦 𝑑 𝑡 2 = 𝑎 𝑦 2 =−𝑅 𝜔 2 𝑠𝑖𝑛𝑡𝜔𝑡

4. R-Form
𝑉 𝑟, 𝜃 = 𝑅𝜔 , 𝑋 𝑟, 𝜃 = 𝑟, 𝜔𝑡

5. Velocity (Derivation)
𝑟= 𝑥 2 + 𝑦 2
= −𝑅𝜔𝑠𝑖𝑛𝜔𝑡 𝑅𝜔𝑐𝑜𝑠𝜔𝑡 2 = 𝑅 2 𝜔 2 𝑠𝑖𝑛 2 𝜔𝑡 + 𝑐𝑜𝑠 2 𝜔𝑡 = 𝑅 2 𝜔 2 =𝑅𝜔

6. Acceleration (Derivation)
𝑟= 𝑥 2 + 𝑦 2 Follow same pathway as with velocity, just use the second derivative taken. 𝑅 2 𝜔 4 𝑐𝑜𝑠 2 𝜔𝑡 + 𝑠𝑖𝑛 2 𝜔𝑡 →𝑅 𝜔 2

7. Overall: 𝑣 = 𝑅𝜔 𝑎 =(𝑅 𝜔 2 ) Fundamental equation of circular motion
Some Conclusions to be made:
𝑣 2 = 𝑅 2 𝜔 2 =𝑅 𝑅 𝜔 2 =𝑅𝑎
𝑎 = 𝑣 2 𝑅
Overall:
𝑣 = 𝑅𝜔
𝑎 =(𝑅 𝜔 2 )
𝑚 𝑠 2
Fundamental equation of circular motion

8. Centripetal vs. Centrifugal

9. Centripetal vs. Centrifugal
Acceleration is always to the center
It is perpendicular to the motion
When this is happening, this is uniform circular motion
CENTRIPETAL MOTION/FORCE
The opposite: centrifugal

10. So why don’t the people fall out of the boat?
Centripetal force.
Inertial or non-accelerational reference frame
Psuedo force
Accelerating in the opposite direction from what you feel the force in
Acceleration are in the same inertial reference frame