1. Chapter 5:Using Newton’s Laws: Friction, Circular Motion, Drag Forces
Chapter Opener. Caption: Newton’s laws are fundamental in physics. These photos show two situations of using Newton’s laws which involve some new elements in addition to those discussed in the previous Chapter. The downhill skier illustrates friction on an incline, although at this moment she is not touching the snow, and so is retarded only by air resistance which is a velocity-dependent force (an optional topic in this Chapter). The people on the rotating amusement park ride below illustrate the dynamics of circular motion.

2. 5-1 Applications of Newton’s Laws Involving Friction
Example 5-7: A ramp, a pulley, and two boxes.
Box A, of mass 10.0 kg, rests on a surface inclined at 37 to the horizontal. It is connected by a lightweight cord, which passes over a massless and frictionless pulley, to a second box B, which hangs freely as shown. (a) If the coefficient of static friction is 0.40, determine what range of values for mass B will keep the system at rest. (b) If the coefficient of kinetic friction is 0.30, and mB = 10.0 kg, determine the acceleration of the system.
Figure 5-9. Caption: Example 5–7. Note choice of x and y axes.
Solution: Pick axes along and perpendicular to the slope. Box A has no movement perpendicular to the slope; box B has no horizontal movement. The accelerations of both boxes, and the tension in the cord, are the same. Solve the resulting two equations for these two unknowns.
The mass of B has to be between 2.8 and 9.2 kg (so A doesn’t slide either up or down).
0.78 m/s2

3. Uniform Circular Motion—Kinematics
Uniform circular motion: motion in a circle of constant radius at constant speed
Instantaneous velocity is always tangent to the circle.
Figure Caption: A small object moving in a circle, showing how the velocity changes. At each point, the instantaneous velocity is in a direction tangent to the circular path.

4. Linear vs. Circular motion:
Linear motion
Relationship
Angular motion
Position
Angle 
Displacement
Angular displacement
Average velocity
Average angular velocity
Instantaneous velocity
Instantaneous angular velocity

5. Uniform circular motion
Uniform circular motion means: Constant rotation speed
How far?
Angular displacement
The size of the angle swept out by the motion
Typically “+” indicates counter-clockwise
Units–radians
Go around once: y  i f r x
2π radians
1 revolution
360 degrees

6. Uniform circular motion
How fast?
Angular velocity  y i f r
This is constant for Uniform circular motion
Units: rad/sec x
rad/s
rpm-revolutions per minute

7. Relationship between angular and linear motion
How far does it go?
Angular displacement,  to linear motion, s.
Here r is the radius of the circle in meters, and s is the distance traveled in meters (or arc length).
 is the angular displacement in radians
since s/r is unitless, s y  i f r x
radians are not a physical unit, and do not need
to balance like most units.

8. Relationship between angular and linear motion
How fast does it go?
Angular velocity , to linear velocity, v
Direction of v is tangent to the circle
Units :
v m/s
 must be in rad/s y v v i f r x

9. Two objects are sitting on a horizontal table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true?
A) Objects 1 and 2 have the same linear velocity, v, and the same angular velocity, .
B) Objects 1 and 2 have the same linear velocity, v, and the different angular velocities, .
C) Objects 1 and 2 have different linear velocities, v, and the same angular velocity, .
D) Objects 1 and 2 have different linear velocities, v, and the different angular velocities, .
Question 1 2

10. Two objects are sitting on a horizontal table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true?
A) Objects 1 and 2 have the same linear velocity, v.
B) Object 1 has a faster linear velocity than object 2.
C) Object 1 has a slower linear velocity than object 2.
Question 1 2

11. Period and frequency
Bring in bicycle wheel

12. Linear vs. Circular motion:
Linear motion
Angular motion
Position
Angle 
Displacement
Angular displacement
Average velocity
Average angular velocity
Instantaneous velocity
Instantaneous angular velocity

13. 5-2 Uniform Circular Motion—Kinematics
Looking at the change in velocity in the limit that the time interval becomes infinitesimally small, we see that
Figure Caption: Determining the change in velocity, Δv, for a particle moving in a circle. The length Δl is the distance along the arc, from A to B.
Velocity can be constant in magnitude, and we
still have acceleration because the direction changes.
Direction: towards the center of the circle

14. Newton’s second law
Whenever we have circular motion, we have acceleration directed towards the center of the motion.
Whenever we have circular motion, there must be a force towards the center of the circle causing the circular motion.
Examples: ball on a string
Planets orbiting sun; moon orbiting earth
Spice on a turntable

15. 5-2 Uniform Circular Motion—Kinematics
This acceleration is called the centripetal, or radial, acceleration, and it points toward the center of the circle.
Figure Caption: For uniform circular motion, a is always perpendicular to v.

16. Uniform Circular Motion—Kinematics
Example 5-10: Ultracentrifuge.
The rotor of an ultracentrifuge rotates at 50,000 rpm (revolutions per minute). A particle at the top of a test tube is 6.00 cm from the rotation axis. Calculate its centripetal acceleration, in “g’s.”
Answer: Again, find v and then a. Convert minutes to seconds and centimeters to meters. Divide final answer by 9.8 m/s2. a = 167,000 g’s.

17. Problem 57
57.(III) The position of a particle moving in the xy plane is given by
where r is in meters and t is in seconds. (a) Show that this represents circular motion of radius 2.0 m centered at the origin. (b) Determine the velocity and acceleration vectors as functions of time. (c) Determine the speed and magnitude of the acceleration. (d) Show that a= v2/r (e) Show that the acceleration vector always points toward the center of the circle.