PHYS 20 LESSONS Unit 5: Circular Motion Gravitation Lesson 2: Dynamics of Circular Motion (Horizontal Circles)

Reading Segment: Dynamics of Uniform Circular Motion To prepare for this section, please read: Unit 5: p.7

Consider an object moving in a circle at a constant speed: What are the directions of the velocity and acceleration vectors?

Consider an object moving in a circle at a constant speed: v a c The velocity vector is tangent to the circle, while the centripetal acceleration vector is directed towards the centre.

v a c What would be the direction of F net ? Why?

v F net a c Based on Newton's 2 nd law, F net must act in the same direction as the acceleration. F net = m a c

v F net = F c a c Since F net is directed towards the centre of the circle as well, then it is called a centripetal force (F c ). This is always true for uniform circular motion.

Centripetal Force For uniform circular motion, the net force is the centripetal force. i.e.F net = F c = m a c What does this mean? - it is directed towards the centre of the circle - it is responsible for turning the object

It must be understood that a centripetal force is not a distinct force. It is only a label, identifying which force(s) is responsible for turning the object.

e.g.If a car is driving in a horizontal (unbanked) circle, what is the centripetal force?

F f = F c a c Friction is the force responsible for keeping the car in a circle.

e.g.A satellite is in uniform circular motion around a planet. What is the centripetal force?

F g = F c The force of gravity holds the satellite in circular motion.

Equations for Centripetal Force Equation 1: Combining the equations F c = m a c and a c = v 2 r we have F c = m v 2 r

Equation 2: Combining the equations F c = m a c and a c = 4  2 r T 2 we have F c = m 4  2 r T 2

Ex. 1A friend attempts to explain a physical phenomenon: "When a car turns sharply, the passenger is pushed into the door with a centrifugal (outward) force." What is wrong with this reasoning? Read Ladner p. 136 Heath p. 191

Very important: There is no such thing as a centrifugal force. Even though you may read about this force in books and see it in internet animations / applets, it does not exist. We call it an "apparent force", because it seems to be acting on us.

So, why do we feel pushed into the door when we go quickly around a corner?

v When the car is forced around the corner by friction, the person wants to travel in a straight line. This is due to inertia (Newton's 1st law) There is no force that pushes us outward.

v F door As a result, the person hits the car. If the person hits the car, then from Newton's 3rd law, we know that the car will exert an equal and opposite reaction force.

v F door This reaction force from the door is the centripetal force that pushes the person around the corner.

Ex. 2A 76.0 kg person is whirled in a horizontal circle of diameter 9.00 m. If the centripetal force on the person is 5.21 kN, then find the time for 1 revolution. See Heath, p. 189 for a diagram of a human centrifuge.

List: m = 76.0 kg r = d = 9.00 m = 4.50 m 2 2 F c = 5.21  10 3 N = 5210 N T ? Equation: F c = m 4  2 r T 2

F c = m 4  2 r T 2 F c T 2 = m 4  2 r T 2 = m 4  2 r F c T = m 4  2 r = (76.0 kg) (4  2 ) (4.50 m) F c 5210 N = 1.61 s

Practice Problems Try these problems in the Physics 20 Workbook: Unit 5 p. 8 #1 - 3, 7

Ex. 3What is the maximum speed that a 700 kg car can round a corner of radius 100 m if the coefficient of friction between the tires and the road is 0.50 ?

Force Diagram: The best force diagram is a 3-D diagram which allows you to draw both vertical and horizontal forces.

Force Diagram: F g F f a c F N The force of friction is the centripetal force.

Vertical forces: F g 1st Law: (balanced forces) F N = F g = m g F N = (700 kg) (9.81 N/kg) = 6867 N

Find friction: F f =  F N = (0.50) (6867 N) = N

Horizontal forces: F f a c 2nd law: F net = m a F f = m a c (or F f = F c )

F net = m a F f = m a c or F f = F c F f = m v 2 r You can choose either approach: Newton's 2nd law or Centripetal force

F f = m v 2 r F f r = m v 2 v 2 = F f r m v = F f r = ( N) (100 m) m700 kg = 22 m/s

Animations Dynamics of Circular Motion: html Dynamics of a Carousel:

Practice Problems Try these problems in the Physics 20 Workbook: Unit 5 p. 8 #4 - 6