Uniform Circular Motion

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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

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Period and Speed• Often easier to use

period T= time to complete 1 revolution instead of linear speed

• Circle=2r • So if v=d/t then V= 2r/TREMEMBER: speed may

be constant but velocity is not! Acceleration changes the direction.

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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

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Centripetal Acceleration and Force

• ac=v2/r and points in

• Fc=macdue to Newton’s 2nd law

• Sometimes written by replacing a so:

Fc=mv2/r

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What provides Fc?

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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!

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FRICTION!

Now solve…

• What is the centripetal force?

• What would happen if the radius of the curve were smaller?

• Fc=mv2/r

• Mass=Fw/g

Fc=400N

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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?

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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

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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 Fc to gravitational force

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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

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Kepler’s 3rd Law

• Remember Newton’s Universal Gravitation, G?

• Kepler equated the force of G with the laws of circular motion to get:

T2/R3 is a constant =42/GMWhere T is period, M is mass of sun, R is

radius of circular orbit (even though it’s not quite circular)