1. Circular Motion Lecturer: Professor Stephen T. Thornton

2. Reading Quiz: When a car is moving around a banked curve, which of the following statements is most true?
The friction is always up the slope of the banked road.
The friction is always down the slope of the banked road.
The friction can either be up or down the slope of the banked road.
It sounds as if the car’s tires are too bald to matter.

3. Answer: C
Friction can be up or down the road. This is in many textbooks and is worked out. There is even one speed where there is no friction.

4. Last Time
Review
Friction

5. Circular motion Centripetal motion, force Motion on banked curves
Today
Circular motion
Centripetal motion, force
Motion on banked curves

7. Try this with a block of wood
Try this with a block of wood. This is similar to the car moving on banked curve.

8. Conceptual Quiz
Below you see two cases: a father pulling or pushing a sled with a force F that is applied at an angle q. In which case is the normal force greater?
A) case 1
B) case 2
C) it’s the same for both
D) depends on the magnitude of the force F
E) depends on the ice surface
Case 1
Case 2
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9. Conceptual Quiz
Below you see two cases: a father pulling or pushing a sled with a force F that is applied at an angle q. In which case is the normal force greater?
A) case 1
B) case 2
C) it’s the same for both
D) depends on the magnitude of the force F
E) depends on the ice surface
Case 1
Case 2
In case 1, the force F is pushing down (in addition to mg), so the normal force needs to be larger. In case 2, the force F is pulling up, against gravity, so the normal force is lessened.

10. Conceptual Quiz
Two blocks of masses 2m and m are in contact on a horizontal frictionless surface. If a force F is applied to mass 2m, what is the force on mass m ? The blocks accelerate.
A) 2F
B) F
C) F
D) F
E) F 2m m F
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11. Conceptual Quiz
Two blocks of masses 2m and m are in contact on a horizontal frictionless surface. If a force F is applied to mass 2m, what is the force on mass m ? The blocks accelerate.
A) 2F
B) F
C) F
D) F
E) F
The force F leads to a specific acceleration of the entire system. In order for mass m to accelerate at the same rate, the force on it must be smaller! How small?? Let’s see... 2m m F
F = 3ma; a = F/3m, F1=ma=F/3
Follow-up: What is the acceleration of each mass?

12. Conceptual Quiz A) the force from the rushing air pushed it off
A box sits in a pickup truck on a frictionless truck bed. When the truck accelerates forward, the box slides off the back of the truck because:
A) the force from the rushing air pushed it off
B) the force of friction pushed it off
C) no net force acted on the box
D) truck went into reverse by accident
E) none of the above
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13. Conceptual Quiz A) the force from the rushing air pushed it off
A box sits in a pickup truck on a frictionless truck bed. When the truck accelerates forward, the box slides off the back of the truck because:
A) the force from the rushing air pushed it off
B) the force of friction pushed it off
C) no net force acted on the box
D) truck went into reverse by accident
E) none of the above
Generally, the reason that the box in the truck bed would move with the truck is due to friction between the box and the bed. If there is no friction, there is no force to push the box along, and it remains at rest. The truck accelerated away, essentially leaving the box behind!!

14. Mass Moving Up Ramp. A small block of mass m is given an initial speed up a ramp inclined at angle to the horizontal. It travels a distance d up the ramp and comes to rest. (a) Determine a formula for the coefficient of kinetic friction between block and ramp. (b) What can you say about the value of the coefficient of static friction?
Giancoli, 4th ed, Problem 5-25

15. Circular motion Do demo with string and ball.
Note that the direction of the velocity is changing. The ball is accelerating!
Do demo with string and ball.

16. Notice that tends to point towards the center of the circle
Notice that tends to point towards the center of the circle. As  becomes smaller and smaller, points directly to center. Therefore the acceleration points towards the center of the circle.

17. Centripetal acceleration
Centripetal means “center seeking”.
Look at your textbook for a derivation of the magnitude of the centripetal acceleration acp :
where r is the radius and v is the speed. r

18. Dynamics of Uniform Circular Motion
We can see that the force must be inward by thinking about a ball on a string. Strings only pull; they never push.
Figure Caption: Swinging a ball on the end of a string.

19. For an object to be in uniform circular motion, there is an acceleration, and, therefore, a net force acting on it.
We already know the acceleration, so we can immediately write the force:
Figure Caption: A force is required to keep an object moving in a circle. If the speed is constant, the force is directed toward the circle’s center.

20. Centripetal force
Where in the world did this centripetal force come from?
There has to be a force to keep the object moving in a circle. In the case of the ball and string, it is the tension in the string. The tension always points towards the center!
The direction of the centripetal force must also be towards the center!

21. The moon rotates around the Earth in a circle
The moon rotates around the Earth in a circle. What is the centripetal force that causes this? If you drive around in a circle with a bicycle or even with a car, what is the centripetal force? In a simple atomic model of the hydrogen atom, the electron rotates around the proton in a circle. What is the centripetal force?
Gravity, friction, electric force

22. How can we make a bowling ball go around in a circle?
How are you going to do it?
What is the centripetal force?
Do on floor in front of side screen. Set up camera to look at it.

23. Circular motion Results for circular motion:
Consider an object moving in a circle of radius r with a constant speed v.
A centripetal acceleration of magnitude v2/r must be present.
There must be a centripetal force Fcp of value

24. Conceptual Quiz: What other forces are exerted on the ball besides mg?
A) Friction
B) Tension
C) A normal force
perpendicular to mg.
D) A normal force
perpendicular to the
surface of the cone at
the ball.

25. Answer: D
The only other possible force is the normal force, and it must be perpendicular to the surface that the ball is rolling upon.

26. Conceptual Quiz: What is the direction of the net force?
A) towards the center of
the dashed circle at the
ball (radially).
B) away from the center
of the circle at the ball.
C) up at the ball.
D) down at the ball.
E) cannot tell with
information given.

27. Answer: A
Because the ball is moving at constant speed in a circle, the net force must be along the radial direction, towards the center of the circle. This is the centripetal force.

28. Look at ball moving in vertical circle
In this case we do not have uniform circular motion. The tension always points towards the center, but gravity points down.
Look at Examples in textbook.
Spin ball in vertical plane. At some point the ball barely makes it around the top, but it has no difficulty swinging around the bottom.

29. Turning corners What do we notice when we turn corners at high speed?
Good chance of falling
over or skidding.
When skiing, we lean
over and tilt our skis.
Interstate highways
are banked.
Motorcycles tilt.

30. Highway Curves: Banked and Unbanked
When a car goes around a curve, there must be a net force toward the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction.
Figure Caption: The road exerts an inward force on a car (friction against the tires) to make it move in a circle. The car exerts an inward force on the passenger.

31. If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show.
Figure Caption: Race car heading into a curve. From the tire marks we see that most cars experienced a sufficient friction force to give them the needed centripetal acceleration for rounding the curve safely. But, we also see tire tracks of cars on which there was not sufficient force—and which unfortunately followed more nearly straight-line paths.

32. You really do not want your tires to slip!!
As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways:
The kinetic frictional force is smaller than the static.
The static frictional force can point toward the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve.
Draw on Elmo.

33. Banking the curve can help keep cars from skidding
Banking the curve can help keep cars from skidding. In fact, for every banked curve, there is one speed at which the entire centripetal force is supplied by the horizontal component of the normal force, and no friction is required. This occurs when:
Figure Caption: Normal force on a car rounding a banked curve, resolved into its horizontal and vertical components. The centripetal acceleration is horizontal (not parallel to the sloping road). The friction force on the tires, not shown, could point up or down along the slope, depending on the car’s speed. The friction force will be zero for one particular speed.

34. Whirling Bucket. A bucket of mass 2
Whirling Bucket. A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N. (a) Find the speed of the bucket. (b) How fast must the bucket move at the top of the circle so that the rope does not go slack?
Giancoli, 4th ed, Problem 5-44

35. Conceptual Quiz A) mk > ms so sliding friction is better
Antilock brakes keep the car wheels from locking and skidding during a sudden stop. Why does this help slow the car down?
A) mk > ms so sliding friction is better
B) mk > ms so static friction is better
C) ms > mk so sliding friction is better
D) ms > mk so static friction is better
E) none of the above
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36. Conceptual Quiz A) mk > ms so sliding friction is better
Antilock brakes keep the car wheels from locking and skidding during a sudden stop. Why does this help slow the car down?
A) mk > ms so sliding friction is better
B) mk > ms so static friction is better
C) ms > mk so sliding friction is better
D) ms > mk so static friction is better
E) none of the above
Static friction is greater than sliding friction, so by keeping the wheels from skidding, the static friction force will help slow the car down more efficiently than the sliding friction that occurs during a skid.

37. Conceptual Quiz Normal Net Force Weight
A box sits on a flat board. You lift one end of the board, making an angle with the floor. As you increase the angle, the box will eventually begin to slide down. Why?
A) component of the gravity force parallel to the plane increased
B) coefficient of static friction decreased
C) normal force exerted by the board decreased
D) both A and C
E) all of A, B, and C
Net Force
Normal
Weight
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38. Conceptual Quiz Normal Net Force Weight
A) component of the gravity force parallel to the plane increased
B) coefficient of static friction decreased
C) normal force exerted by the board decreased
D) both A and C
E) all of A, B, and C
A box sits on a flat board. You lift one end of the board, making an angle with the floor. As you increase the angle, the box will eventually begin to slide down. Why?
As the angle increases, the component of weight parallel to the plane increases and the component perpendicular to the plane decreases (and so does the normal force). Because friction depends on normal force, we see that the friction force gets smaller and the force pulling the box down the plane gets bigger.
Net Force
Normal
Weight

39. Conceptual Quiz
A mass m is placed on an inclined plane (m > 0) and slides down the plane with constant speed. If a similar block (same m) of mass 2m were placed on the same incline, it would:
A) not move at all
B) slide a bit, slow down, then stop
C) accelerate down the incline
D) slide down at constant speed
E) slide up at constant speed m
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40. Conceptual Quiz
A mass m is placed on an inclined plane (m > 0) and slides down the plane with constant speed. If a similar block (same m) of mass 2m were placed on the same incline, it would:
A) not move at all
B) slide a bit, slow down, then stop
C) accelerate down the incline
D) slide down at constant speed
E) slide up at constant speed q W N f Wx Wy
The component of gravity acting down the plane is double for 2m. However, the normal force (and hence the friction force) is also double (the same factor!). This means the two forces still cancel to give a net force of zero.