1. 7.1 Uniform Circular Motion and & 7.2 Gravitation
Chapter 7
7.1 Uniform Circular Motion and & 7.2 Gravitation

2. Uniform Circular Motion
An object that moves at uniform speed in a circle of constant radius is said to be in uniform circular motion.
Question: Why is uniform circular motion accelerated motion?
Answer: Although the speed is constant, the velocity is not since an object in uniform circular motion is continually changing direction.

3. Centrifugal Force Question: What is centrifugal force?
Answer: That’s easy. Centrifugal force is the force that flings an object in circular motion outward. Right?
Wrong! Centrifugal force is a myth! There is no outward directed force in circular motion. To explain why this is the case, let’s review Newton’s 1st Law.

4. Newton’s 1st Law and cars
When a car accelerates forward suddenly, you as a passenger feel as if you are flung backward.
You are in fact NOT flung backward. Your body’s inertia resists acceleration and wants to remain at rest as the car accelerates forward.
When a car brakes suddenly, you as a passenger feel as if you are flung forward.
You are NOT flung forward. Your body’s inertia resists acceleration and wants to remain at constant velocity as the car decelerates.

5. When a car turns You feel as if you are flung to the outside.
You are NOT flung to the outside. Your inertia resists the inward acceleration and your body simply wants to keep moving in straight line motion!
As with all other types of acceleration, your body feels as if it is being flung in the opposite direction of the actual acceleration. The force on your body, and the resulting acceleration, actually point inward.

6. Centripetal Acceleration
Centripetal (or center-seeking) acceleration points toward the center of the circle.
This type of acceleration is at right angles to the velocity.

7. Centripetal Acceleration
ac = v2/r
ac: centripetal acceleration in m/s2
v: tangential speed in m/s
r: radius in meters v ac

8. Centripetal Force Fc = m ac Fc = m v2 / r
A force responsible for centripetal acceleration is referred to as a centripetal force.
Newton’s second law states,
Fc = m ac
Fc = m v2 / r
Fc: centripetal force in N
v: tangential speed in m/s
r: radius in meters Fc
Always toward center of circle!

9. Any force can be centripetal
The name “centripetal” can be applied to any force in situations when that force is causing an object to move in a circle.
You can identify the real force or combination of forces which are causing the centripetal acceleration.

10. Static friction
As a car makes a turn on a flat road, what is the real identity of the centripetal force?

11. Tension
As a weight is tied to a string and spun in a circle, what is the real identity of the centripetal force?

12. Gravity
As the moon orbits the Earth, what is the real identity of the centripetal force?

13. Normal force with help from static friction
As a racecar turns on a banked curve on a racing track, what is the real identity of the centripetal force?

14. Tension, with some help from gravity
As you swing a mace in a vertical circle, what is the true identity of the centripetal force?

15. Gravity, with some help from the normal force
When you are riding the Tennessee Tornado at Dollywood, what is the real identity of the centripetal force when you are on a vertical loop?

16. Newton’s Law of Universal Gravitation
Every object in the universe exerts a gravitational attraction to all other objects in the universe
The amount of gravitational force depends upon the mass of the objects and the distance between the objects

17. The strength of gravity
The attraction is directly proportional to the product of their masses
The attraction is inversely proportional to the square of the distance between them (1/d2).
The law of universal gravitation can be expressed as an exact equation when a proportionality constant is introduced.
The universal gravitational constant, G, in the equation for universal gravitation describes the strength of gravity.

18. Newton’s Law of Universal Gravitation

19. Weight Revisited weight = mass  gravitational field strength
Because it depends on gravitational field strength, weight changes with location:
On the surface of any planet, the value of g, as well as your weight, will depend on the planet’s mass and radius.

20. Sample problem
A 1200-kg car rounds a corner of radius r = 45 m. If the coefficient of static friction between tires and the road is 0.93 and the coefficient of kinetic friction between tires and the road is 0.75, what is the maximum velocity the car can have without skidding?

21. Sample problem solutionmg N fs
SF = ma
fs = mv2/r
msN = mv2/r
msmg = mv2/r
v = (msgr)1/2
v = [(.93)(9.8)(45)]1/2
v = 20.2 m/s

22. Sample problem
You whirl a 2.0 kg stone in a horizontal circle about your head. The rope attached to the stone is 1.5 m long.
a) What is the tension in the rope? (The rope makes a 10o angle with the horizontal).
b) How fast is the stone moving?

23. Sample solution – (a) SFy = 0 Tsin10o – mg = 0 T = mg/sin10o
T = 113 N mg T
10o
Tcos10o
Tsin10o

24. Sample solution – (b) SFx = ma Tcos10o = mv2/rmg T
10o
Tcos10o
Tsin10o
SFx = ma
Tcos10o = mv2/r
v = [(Tcos10o)(r)/m)]1/2
T comes from previous problem
r comes from length of rope and angle at which rope is angled
v = [(113)(cos10o)(1.47)/2.0]1/2
v = 9.0 m/s
1.5 m
10o
r = (1.5m)(cos 10o)
r = 1.47 m