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. 𝑣 𝑟

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 = 𝑣 2 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? Δs 𝜃

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

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

12. Centripetal Acceleration - Example
A centrifuge creates a centripetal acceleration of 𝑚 𝑠 2 . 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 =61250 𝑚 𝑠 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. 𝑣 = ? 𝑟

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. 𝑣 = ? 𝑟 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 = 𝐺 𝑚 1 𝑚 2 𝑟 2
𝐺 is the universal gravitational constant:
𝐺= −11 𝑁 𝑚 2 𝑘 𝑔 2

23. Newton’s Law of Universal Gravity
𝐹 𝐺 = 𝐺 𝑚 1 𝑚 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
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 =0.5 𝑅 𝐸 𝑀 𝑝 =0.1 𝑀 𝐸 𝑔 𝑝 =?

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= −11 𝑁 𝑚 2 𝑘 𝑔 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 𝑅 𝐴 =2 𝑅 𝐸 𝑇 𝐴 = ?