Uniform Circular Motion

What is uniform circular motion? Constant speed Circular path Must be an unbalanced force acting towards axis of rotation- think free body diagrams! Ex of forces: tension, banked curves, gravitation

Period and Speed Often easier to use period T= time to complete 1 revolution instead of linear speed Circle=2  r So if v=d/t then V= 2  r/T REMEMBER: speed may be constant but velocity is not! Acceleration changes the direction.

Vectors in circular motion Velocity points tangent to circle Acceleration points in to axis of rotation because a=  v/  t and  v is always towards center

Centripetal Acceleration and Force a c =v 2 /r and points in F c =ma c due to Newton’s 2nd law Sometimes written by replacing a so: F c =mv 2 /r

What provides F c ?

DRAW Free body diagrams Ex: An athlete who weights 800N is running around a curve at a speed of 5.0m/s in an arc whose radius is 5.0m. What provides the centripetal force? Draw a free body diagram! FRICTION!

Now solve… What is the centripetal force? What would happen if the radius of the curve were smaller? F c =mv 2 /r Mass=F w /g F c =400N

Now take it 2 step further… If the coefficient of static friction btwn the shoe and the track =1 then will the runner slip? How does changing the radius of the curve affect whether the runner will slip?

Another example A roller coaster enter as loop. At the very top the speed of the car is 25m/s and the acceleration points straight down. If the diameter of the loop is 50m and the total mass of the car=1200kg, what is the magnitude of the normal force? Start with a free body diagram- what forces are acting? If net force is straight down, why doesn’t the car fall off the track?

Banked Curves Draw a free body diagram for a car traveling around a banked curve- even without friction Nsin  is component of force keeping car on curve- even without any friction.

Circular motion and universal gravitation Satellites, planets, moons, etc can travel in circular paths- to solve, equate F c to gravitational force

Kepler’s Laws: 1 and 2 Every planet moves in elliptical orbit with sun at 1 focus As planet moves in its orbit, a line drawn from sun to planet sweeps out equal area in equal time

Kepler’s 3rd Law Remember Newton’s Universal Gravitation, G? Kepler equated the force of G with the laws of circular motion to get: T 2 /R 3 is a constant =4  2 /GM Where T is period, M is mass of sun, R is radius of circular orbit (even though it’s not quite circular)