1. Circular Motion and Other Applications of Newton’s Laws
Chapter 6
Circular Motion
and
Other Applications of Newton’s Laws

2. Uniform Circular Motion
A force, Fr , is directed toward the center of the circle
This force is associated with an acceleration, ac
Applying Newton’s Second Law along the radial direction gives

3. Uniform Circular Motion, cont
A force causing a centripetal acceleration acts toward the center of the circle
It causes a change in the direction of the velocity vector
If the force vanishes, the object would move in a straight-line path tangent to the circle

4. Centripetal Force
The force causing the centripetal acceleration is sometimes called the centripetal force
This is not a new force, it is a new role for a force
It is a force acting in the role of a force that causes a circular motion

5. Conical Pendulum
The object is in equilibrium in the vertical direction and undergoes uniform circular motion in the horizontal direction
v is independent of m

6. Motion in a Horizontal Circle
The speed at which the object moves depends on the mass of the object and the tension in the cord
The centripetal force is supplied by the tension

7. Horizontal (Flat) Curve
The force of static friction supplies the centripetal force
The maximum speed at which the car can negotiate the curve is
Note, this does not depend on the mass of the car

8. Banked Curve These are designed with friction equaling zero
There is a component of the normal force that supplies the centripetal force

9. Loop-the-Loop This is an example of a vertical circle
At the bottom of the loop (b), the upward force experienced by the object is greater than its weight

10. Loop-the-Loop, Part 2
At the top of the circle (c), the force exerted on the object is less than its weight

11. Non-Uniform Circular Motion
The acceleration and force have tangential components
Fr produces the centripetal acceleration
Ft produces the tangential acceleration
SF = SFr + SFt

12. Vertical Circle with Non-Uniform Speed
The gravitational force exerts a tangential force on the object
Look at the components of Fg
The tension at any point can be found

13. Top and Bottom of Circle
The tension at the bottom is a maximum
The tension at the top is a minimum
If Ttop = 0, then

14. Motion in Accelerated Frames
A fictitious force results from an accelerated frame of reference
A fictitious force appears to act on an object in the same way as a real force, but you cannot identify a second object for the fictitious force

15. “Centrifugal” Force
From the frame of the passenger (b), a force appears to push her toward the door
From the frame of the Earth, the car applies a leftward force on the passenger
The outward force is often called a centrifugal force
It is a fictitious force due to the acceleration associated with the car’s change in direction

16. Uniform Circular Motion
Uniform circular motion occurs when an object moves in a circular path with a constant speed
An acceleration exists since the direction of the motion is changing
This change in velocity is related to an acceleration
The velocity vector is always tangent to the path of the object

17. Changing Velocity in Uniform Circular Motion
The change in the velocity vector is due to the change in direction
The vector diagram shows Dv = vf - vi

18. Centripetal Acceleration
The acceleration is always perpendicular to the path of the motion
The acceleration always points toward the center of the circle of motion
This acceleration is called the centripetal acceleration

19. Centripetal Acceleration, cont
The magnitude of the centripetal acceleration vector is given by
The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

20. Period The period, T, is the time required for one complete revolution
The speed of the particle would be the circumference of the circle of motion divided by the period
Therefore, the period is

21. Tangential Acceleration
The magnitude of the velocity could also be changing
In this case, there would be a tangential acceleration

22. Total Acceleration
The tangential acceleration causes the change in the speed of the particle
The radial acceleration comes from a change in the direction of the velocity vector

23. Total Acceleration, equations
The tangential acceleration:
The radial acceleration:
The total acceleration:
Magnitude