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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)