Uniform Circular Motion

Transcription

1 Circular Motion

2 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

3 r v

4 v =? r f Δθ r 0

5 Δ v v f v 0 r f v 0 Δθ r 0

6 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

7 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

8 Period, Frequency, & Angular Frequency Period and frequency are related by: f = 1 T There are 2π radians in one rotation, therefore: ω = 2π f

9 Angular Frequency Equations If an object rotates through an angle θ with a radius r, how far does the object travel? θ Δs

10 Angular Frequency Equations In general: Δs = r θ and v = r ω

11 Angular Frequency Equations Combining a c = v2 r and v = r ω gives: a c = r ω 2

12 Centripetal Acceleration - Example A centrifuge creates a centripetal acceleration of 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?

13 a c = m s 2 v =? r = 0.05 m f =?

14 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?

15 r v =?

16 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?

17 r v =? θ

18 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.

19 Circular Motion with Gravity

20 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?

21 r = 4.0 m

22 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 = N m2 kg 2

23 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

24 Using Newton s Law of Universal Gravity, we can show that g = 9.80 m/s 2. M E = kg R E = m Using this formula, we can calculate the acceleration due to gravity on planets other than Earth.

25 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?

26 R p = 0.5R E g p =? M p = 0.1M E

27 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

28 Universal Gravity - Example How fast is the satellite moving when it is placed in a circular orbit around the Earth with a radius of m? The Earth has a mass of kg.

29 r = m M E = kg G = N m2 kg 2 v =?

30 Kepler s Third Law of Planetary Motion Kepler s Third Law relates the period T to the radius of the orbit R

31 Kepler s Third Law of Planetary Motion Kepler s Third Law of Planetary Motion Says: T 2 R 3 = const.

32 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?

33 Kepler s Third Law - Example T E = 1 year T A =? R A = 2 R E