1. Circular Motion

2. Rotating Turning about an internal axis Revolving Turning about an external axis

3. Linear speed, v How far you go in a certain amount of time Miles per hour, meters per second Rotational speed,  How many times you go around in a certain amount of time Revolutions per minute, rotations per hour, radians per second  = v/r (r = radius)

4. Which horse has a larger linear speed on a merry go round, one on the outside or one on the inside? Outside. Which horse has a greater rotational speed? Neither, all the horses complete the circle in the same amount of time.

5. How much faster will a horse at TWICE the distance from the center of the circle be moving Since both horses complete a circle in the same time, they have the same rotational speed, .  = v/r v =  r TWICE the distance means TWICE the speed

6. How do you find the velocity if it is not directly provided? Velocity = distance / time In circular motion, the distance traveled is all around the circle… the circumference. The circumference = 2  r So… v = 2  r / t

7. The number of revolutions per second is called the frequency, f. Frequency is measured in Hertz, Hz. The time it takes to go all the way around once is called the period, T. Frequency is related to period by f = 1 / T

8. Uniform Circular Motion, UCM: moving in a circle with a constant speed. Question: Is there a constant velocity when an object moves in a circle with a constant speed? No, the direction changes, therefore the velocity changes. If the velocity changed, the object is actually ACCELERATING even while moving at the same speed.

9. Suppose an object was moving in a straight line with some velocity, v. According to Newton’s 1 st Law of Motion, “An object in motion continues that motion unless a net external force acts on it”. If you want the object to move in a circle, some force must push or pull it towards the center of the circle. A force that pushes or pulls an object towards the center of a circle is called a centripetal force Centripetal means “center seeking ”

10. According to Newton’s 2 nd Law,  F = ma, If there is a centripetal force, there must be a centripetal acceleration. a c = v 2 / r Where r is the radius of the circle and v is the velocity of the object.

11. Centripetal force Since  F= ma, the net centripetal force is given by

12. Lots of forces can help in pushing or pulling an object towards (or away from) the center of a circle. Sometimes it takes more than one force to get an object to move in uniform circular motion. Centripetal force is the NET force pointing along the radius of a circle, NOT a new kind of force. If an object moves in a circle (or an arc), there must be at least one force that is acting toward the center of the circle. In a free body diagram, NEVER label an individual force as “Centripetal force”!!

13. When can these forces contribute to a net centripetal force acting on an object that is moving in a circular pathway? Gravity? Moon revolving around the Earth Tension? Twirling a pail at the end of a string Friction? Cars rounding a curve. Air Resistance (“Lift”)? Airplane or birds flying in a circle. Normal? Riders in a carnival ride

16. Example A boy twirls a ½ kg rock in a horizontal circle on the end of a 1.6 meter long string. If the velocity of the rock was 4 m/s, what is the Tension in the string? m = ½ kg r = 1.6 m v = 4 m/s The only centripetal force is Tension. T = m v 2 / r T = ½ 4 2 / 1.6 T = 5 N

17. Example How fast was the ½ kg rock moving if the Tension was 10 N and the string was 1.6 m long? m = ½ kg r = 1.6 m T = 10 N T = mv 2 / r Tr/m = v 2 10 x 1.6 /.5 = v 2 v = 5.7 m/s

18. Only a component of the tension acts as a centripetal force!  F = mv 2 /r Tsin  = mv 2 /r and… Tcos  = mg

19. For training, astronauts are required to ride in a special centrifuge to simulate the great accelerations they will experience during lift- off. If the radius is 10 meters, what velocity must the astronaut move to not exceed the acceleration of free-fall? a = v 2 /r = 9.8 m/s 2 v 2 = 10 m x 9.8 m/s 2

20. “Artificial Gravity” Occupants of a space station feel weightless because they lack a support (Normal) force pushing up against their feet. By spinning the station as just the right speed, they will experience a “simulated gravity” when the Normal force of the floor pushing up on their feet becomes a centripetal force. The closer their centripetal acceleration, v 2 /r is to g, the acceleration due to Earth’s gravity, the more they feel the sensation of normal weight.

21. Artificial Gravity Example… A circular rotating space station has a radius of 40 m. What linear velocity, v, must be maintained along the outer edge, to maintain a sense of “normal” gravity? We want the centripetal acceleration, v 2 /r (due to the rotation) to be the same as the acceleration due to gravity on Earth- 9.8 m/s 2 a centripetal = a gravity v 2 /r = 9.8 m.s 2 v 2 = 9.8 m/s 2 x 40 m v = 19.8 m/s

22. Friction A 1500 kg race car goes around a curve at 45 m/s. If the radius of the curve is 100 m, how much friction is required to keep the car on the track? What is , the coefficient of friction? m = 1500 kg v = 45 m/s r = 100 m The centripetal force is friction. f = mv 2 /r f = 1500 x 45 2 / 100 f = 30375 N f =  N  = f / N N = mg = 15000 N  = 30375 N / 15000 N  = 2.02

23. The Normal force In some cases the normal force can contribute to the net centripetal force. For example, on the carnival ride where the riders stand against the walls of the circular room and the floor drops out! And yet, the rider does not slide down! Draw the free-body diagram! What keeps them from sliding down? The wall pushes against the rider toward the center of the circle. N = m v 2 / r mg f N