2. Uniform Circular Motion
Here is a flow chart showing topics from uniform circular motion which hopefully you will read about and understand during your study.

3. Acceleration Revisited
Acceleration is the time rate of change in velocity of an object.
Any time an object is changing its velocity, it is accelerating.
Anytime object changing speed experiences acceleration, but is that the only way?
SCHWINN; Insert the graphic above minus the the green letters and the mathematical symbols for integration put an X in the blue area and a V for the velocity area. ASK THE QUESTION: but is that the only way? And then say what if you are not going in a strait line direction?
Acceleration Revisited

4. Distinct Ways That A Vector Quantity Can Be Changed
Recall that velocity is a vector quantity. A vector can be changed by
(i) Changing the magnitude of the vector
(ii) Changing the direction of the vector
(iii) Changing both the magnitude and direction of the vector
During a race a car can change speeds and directions. The first part of this course we worried about only changing speeds when we had an acceleration. We will now have to consider when the vector quantity of velocity has a magnitude (speed) and or a directional change. If it has a directional change it will have an acceleration which we will consider to be a “Centripetal” acceleration.

5. Distinct Ways That The Velocity of An Object Can Be Changed
An object will therefore experience an acceleration if (i) The speed of the object can change while it travels in a straight line (ii) The direction of travel of the object can change while it travels at constant speed (iii) The object can change speed and change direction at the same time
An object will therefore experience an acceleration if
(i) The speed of the object can change while it travels in a straight line (ii) The direction of travel of the object can change while it travels at constant speed (iii) The object can change speed and change direction at the same time

6. Uniform Circular Motion
An object moving a constant speed but moving in a circular path is called:
Circular Motion
speed stays the same but direction changes→acceleration is occuring)
Schwinn: you could animate a picture of a car moving around a circular track here. You could show a picture of the speedometer and a compass. The speedometer could have a constant value but the compass would have to change as the direction of the car changes…
Uniform Circular Motion

7. An object moving along a circular path is not traveling in a straight line. This means that you cannot use the straight-line kinematics formulas developed in previous lessons to describe the motion of such an object.
An object moving along a circular path is not traveling in a straight line. This means that you cannot use the straight-line kinematics formulas developed in previous lessons to describe the motion of such an object. Always remember the limitations of the models we develop in order to avoid the pitfalls that many a student before you has fallen into. You must use the equations for circular motion which we are about to demonstrate/derive.
Warning!

8. Motion in Two Dimensions: Circular Motion
For circular motion at a constant speed, the acceleration vector a points toward the center of the circle.
An acceleration that always points directly toward the center of a circle is called a centripetal acceleration.
Centripetal acceleration is just the name for a particular type of motion. It is not a new type of acceleration.
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9. Motion in Two Dimensions: Circular Motion
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10. Motion in Two Dimensions: Circular Motion
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11. Motion in Two Dimensions: Circular Motion
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12. Velocity and Acceleration in Uniform Circular Motion
Although the speed of a particle in uniform circular motion is constant, its velocity is not constant because the direction of the motion is always changing.
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13. The time it takes to go around a circle once is called: Period (T) The distance around a circle is called circumference (c) c=πd or 2πr r v
Note:Constant speed when in UCM→ 𝑣 avg = 𝑣 inst
Period

14. Centripetal Acceleration
One way of measuring the speed of an object moving in a circle is to measure its period, T, the time needed for the object to make one complete revolution.
During this time, the object travels a distance equal to the circumference of the circle, 2πr. The object’s speed, then, is represented by v = 2πr/T.
Section 6.2-6

15. A Quick Summary
An object traveling in a circular path at constant velocity does indeed experience an acceleration. This acceleration is toward the center of the circular path at all times. The velocity vector for the object is always tangent to the circle. NOTE: The expressions from straight line kinematics cannot be applied to the motion of such an object.

16. Centripetal Acceleration
The acceleration of an object moving in a circle is always in the direction of the net force acting on it, there must be a net force toward the center of the circle. This force can be provided by any number of agents.
When an Olympic hammer thrower swings the hammer, the force is the tension in the chain attached to the massive ball.
Section 6.2-7

17. Centripetal Acceleration
When an object moves in a circle, the net force toward the center of the circle is called the centripetal force.
To analyze centripetal acceleration situations accurately, you must identify the agent of the force that causes the acceleration. Then you can apply Newton’s second law for the component in the direction of the acceleration in the following way.
Section 6.2-8

18. Circular motion component form of Newton's second law
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Circular motion component form of Newton's second law
So for UCM ∑Fnet is now :
Σ 𝐹 𝑟 =𝑚 𝑎 𝑟 𝑎𝑛𝑑 𝑎 𝑟 = 𝑣 2 𝑟
𝜮 𝑭 𝒓 =𝒎 𝒗 𝟐 𝒓 vice ma

19. Dynamics of Uniform Circular Motion
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is not a new kind of force: The is due to a force such as tension, friction, or the normal force.
We call this centripetal force (center seeking)
Dynamics of Uniform Circular Motion

20. Question 1
Explain why an object moving in a circle at a constant speed is accelerating.
Section

21. Answer 1
Acceleration is the rate of change of velocity, the object is accelerating due to its constant change in the direction of its motion.
Section

22. Question 2
What is the relationship between the magnitude of centripetal acceleration (ac) and an object’s speed (v)? A. B. C. D.
Section

23. Answer 2 Reason: From the equation for centripetal acceleration:
Centripetal acceleration always points to the center of the circle. Its magnitude is equal to the square of the speed divided by the radius of the motion.
Section

24. Question 3
What is the direction of the velocity vector of an accelerating object?
A. toward the center of the circle
B. away from the center of the circle
C. along the circular path
D. tangent to the circular path
Section

25. Answer 3
Reason: While constantly changing, the velocity vector for an object in uniform circular motion is always tangent to the circle. Vectors are never curved and therefore cannot be along a circular path.
Section

26. © 2015 Pearson Education, Inc.
A ball at the end of a string is being swung in a horizontal circle. What force is producing the centripetal acceleration of the ball?
Gravity
Air resistance
Normal force
Tension in the string
Answer: D
Question 1

27. © 2015 Pearson Education, Inc.
A ball at the end of a string is being swung in a horizontal circle. What force is producing the centripetal acceleration of the ball?
Gravity
Air resistance
Normal force
Tension in the string
Question 1

28. © 2015 Pearson Education, Inc.
A ball at the end of a string is being swung in a horizontal circle. What is the direction of the net force on the ball?
Tangent to the circle
Toward the center of the circle
There is no net force.
Answer: B
Question 2

29. © 2015 Pearson Education, Inc.
A ball at the end of a string is being swung in a horizontal circle. What is the direction of the net force on the ball?
Tangent to the circle
Toward the center of the circle
There is no net force.
Question 2

30. © 2015 Pearson Education, Inc.
A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the coin’s velocity?
Answer: A
Question 3

31. © 2015 Pearson Education, Inc.
A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the coin’s velocity? A
Question 3

32. © 2015 Pearson Education, Inc.
A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the frictional force on the coin?
Answer: D
Question 4

33. © 2015 Pearson Education, Inc.
A coin is rotating on a turntable; it moves without sliding. At the instant shown in the figure, which arrow gives the direction of the frictional force on the coin? D
Question 4

34. © 2015 Pearson Education, Inc.
A coin is rotating on a turntable; it moves without sliding. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move?
Answer: A
Question 5

35. © 2015 Pearson Education, Inc.
A coin is rotating on a turntable; it moves without sliding. At the instant shown, suppose the frictional force disappeared. In what direction would the coin move? A
Question 5