1. Circular Motion and Gravitation

2. Review If an object travels at a constant velocity: 𝑎 = 0 𝐹 𝑛𝑒𝑡 = 0
𝐹 𝑛𝑒𝑡 = 0
If an object travels with changing velocity:
𝑎= 𝑣 𝑓 − 𝑣 𝑜 𝑡
𝐹 𝑛𝑒𝑡 =𝑚𝑎

3. What if?
What if you do not speed up or slow down BUT change direction?
Change in velocity occurs.
Acceleration occurs.
𝐹 𝑛𝑒𝑡 =𝑚𝑎

4. Centripetal Acceleration
‘Centripetal’ means ‘center seeking’
Points towards the center
Sometimes called radial acceleration
Dependent on v and r
𝑎 𝑐 = 𝑣 2 𝑟

5. Express the acceleration in m/s2 and g’s.
Example
A jet plane traveling at 525 m/sec pulls out of a dive by moving in an arc of radius 6 km. What is the plane’s centripetal acceleration?
Express the acceleration in m/s2 and g’s.

7. Example
A ball at the end of string is revolving in a horizontal circle of radius 0.6 m. The ball makes 2 revolutions in a second. What is the centripetal acceleration?
Whoa! We need to figure out the velocity first!
Time per revolution = period (T)… measured in seconds (s)

10. Example – You Try
The Moon’s nearly circular orbit about the Earth has a radius of 384,000 km and a period of days. Determine the acceleration of the Moon toward the Earth.

12. Example – You Try
Taking the age of Earth to be about 4 ✕ 109 years and assuming its orbital radius of 1.5 ✕ 1011 m has not changed and is circular, calculate the approximate total distance Earth has traveled since its birth (in a frame of reference stationary with respect to the Sun).

14. Period and Revolutions
We will often see the terms ‘period’, ‘revolutions’, and ‘revolutions per unit time’
Period – the amount of time to make one rotation (T)
Revolution – one rotation
Revolution per unit time – the inverse of the period (1/T)

15. Dynamics of Circular Motion
If there is an acceleration, Newton’s 2nd Law says there is a net external force.
Since the acceleration in centripetal, the net force must also be centripetal.
𝐹 𝑛𝑒𝑡 =𝑚𝑎
𝐹 𝑐 = 𝑚 𝑣 2 𝑟
𝐹 𝑐 =𝑚 𝑎 𝑐

16. Centripetal Force
But that would mean centripetal force is not a real force, but is a net force and caused by something else.
Fn, Fg, Ff, Fp…
It is the sum of the forces acting on the radial axis.

17. That “other” Force
Centrifugal - an outward force acting on an object moving in a circular path.
Everyone has “felt” it.
So, what do you feel? Depends on who “you” are.

18. “Centrifugal Force”
If the centrifugal force really existed what would happen to the ball if you let go?

19. Example
A 10 kg mass is attached to a string that has a breaking strength of 200 N. If the mass is whirled in a horizontal circle of radius 0.8 m, what maximum speed can it have?

21. Example
A flat puck of mass M is rotated in a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block of mass m. What is the speed of the flat puck if the radius of the circle is R?

23. Example – You Try
A typical red blood cell has a mass of 3.0 ✕ kg. A blood sample placed in a centrifuge is subjected to a force of 5.0✕10-11 N when the centrifuge is operated at 140 rev/s. What is the diameter of the centrifuge?

25. Circular Motion and Friction
The centripetal force is provided for by friction.
BUT… friction (static… no sliding) has a maximum value
There is a maximum speed you can take a turn.

26. Example
A 1000 kg car rounds a curve on a flat road of radius 50 m at a speed of 15 m/sec. What minimum coefficient of friction is needed for the car to safely navigate the corner?

28. Example – You Try
An unbanked (flat) curve of radius 160 m is rated for a maximum speed of 45.0 m/s. At what maximum speed should a flat curve with radius of m be rated?

30. Warm-Up
A child has a toy tied to the end of a string and whirls the toy at constant speed in a horizontal circular path of radius R. The toy completes each revolution of its motion in a time period T. What is the magnitude of the acceleration of the toy?

32. Agenda Circular Motion w/ Angles Tomorrow: Circular Motion Lab So far…
Identify the direction of the velocity, acceleration, and force vectors of an object in circular motion.
Determine the velocity, acceleration, and centripetal force acting on an object moving in a horizontal circle.
Today…
Identify the forces (components of forces) acting along the radial axis.
Determine centripetal force and other unknowns of objects moving in circular orbits with forces at angles.

33. Circular Motion and Angles!
Often times the force(s) providing the centripetal force must also balance other forces (mostly gravity)
Since there is a net force acting, we have to figure out which forces are on the radial axis and solve.

34. Example
A 300 g tetherball is attached to a massless rope 2 m long. If the rope makes an angle of 30° to the vertical, what is the speed of the ball?

37. Example
A 4 kg object is attached to a vertical rod by two strings. The object rotates in a horizontal circle at constant speed 6 m/sec. Find the tension in each of the strings.

39. Example – You Try
In a version of a “Giant Swing”, the seat is connected to two cables as shown. The seat swings in a horizontal circle at a rate of 15 rpm. If the seat weighs 255 N and a 825 N person is sitting in it, find the tension in each cable.