Circular Motion
Uniform Circular Motion Uniform Circular Motion Traveling with a constant speed in a circular path Even though the speed is constant, the acceleration is non-zero The acceleration responsible for uniform circular motion is called centripetal acceleration We can calculate a c by relating Δ v to Δt
r v
v =? r f Δθ r 0
Δ v v f v 0 r f v 0 Δθ r 0
Centripetal Acceleration When an object travels with uniform circular motion, the acceleration always points towards the center of the circular path with: a c = v2 r
Period, Frequency, & Angular Frequency Period (T) The time it takes to complete one full rotation Frequency (f) Rotations per second (or per minute) Angular Frequency (ω) Radians per second
Period, Frequency, & Angular Frequency Period and frequency are related by: f = 1 T There are 2π radians in one rotation, therefore: ω = 2π f
Angular Frequency Equations If an object rotates through an angle θ with a radius r, how far does the object travel? θ Δs
Angular Frequency Equations In general: Δs = r θ and v = r ω
Angular Frequency Equations Combining a c = v2 r and v = r ω gives: a c = r ω 2
Centripetal Acceleration - Example A centrifuge creates a centripetal acceleration of 61250 m s2. The average radius of the arm of the centrifuge is r = 5 cm. How fast does the centrifuge spin in revolutions per second?
a c = 61250 m s 2 v =? r = 0.05 m f =?
Centripetal Acceleration - Example A car drives around a level turn. The tires have a coefficient of friction of μ s = 0.8, and the turn has a radius of 90 m. How fast can the car go around the turn without sliding?
r v =?
Centripetal Acceleration - Example A car drives around an banked turn with a radius of 90 m. The turn is designed so that a car traveling 10 m/s will be able go around the turn even when the coefficient of friction is reduced to μ s = 0. What angle is the turn banked at?
r v =? θ
Circular Motion with Gravity The centripetal force is any force that holds an object in circular motion. That is, any force that points inwards towards the center of a circular path. As an object goes around a loop, the forces that make up the centripetal force change.
Circular Motion with Gravity
Circular Motion with Gravity How fast does a rollercoaster need to be traveling when it goes through a vertical loop with a radius of 4.0 m?
r = 4.0 m
Newton s Law of Universal Gravity Every particle exerts an attractive force all other particles The force is given by: F G = Gm 1m 2 r 2 G is the universal gravitational constant: G = 6.67 10 11 N m2 kg 2
Newton s Law of Universal Gravity F G = Gm 1m 2 r 2 Note that r, is the distance between the centers of the two masses Therefore, when standing on the surface of the Earth: r = R E + h R E
Using Newton s Law of Universal Gravity, we can show that g = 9.80 m/s 2. M E = 5.97 10 24 kg R E = 6.37 10 6 m Using this formula, we can calculate the acceleration due to gravity on planets other than Earth.
Universal Gravity - Example Another planet is discovered. The new planet has a radius half the radius of the Earth, R p = 0.5 R E, and one-tenth the mass of Earth M p = 0.1 M E. What is the acceleration due to gravity on this planet?
R p = 0.5R E g p =? M p = 0.1M E
Orbit Any force can supply the centripetal force required to keep an object in uniform circular motion. When a satellite obits the Earth, the centripetal force is supplied by the gravitational force. F g
Universal Gravity - Example How fast is the satellite moving when it is placed in a circular orbit around the Earth with a radius of 3.59 10 7 m? The Earth has a mass of 5.97 10 24 kg.
r = 3.59 10 7 m M E = 5.97 10 24 kg G = 6.67 10 11 N m2 kg 2 v =?
Kepler s Third Law of Planetary Motion Kepler s Third Law relates the period T to the radius of the orbit R
Kepler s Third Law of Planetary Motion Kepler s Third Law of Planetary Motion Says: T 2 R 3 = const.
Kepler s Third Law - Example An asteroid orbits the Sun with a radius that is exactly twice the radius of the Earth s orbit around the Sun. How long does it take this asteroid to orbit the Sun?
Kepler s Third Law - Example T E = 1 year T A =? R A = 2 R E