1. More Applications of Newton’s Laws
Chapter 5
More Applications of
Newton’s Laws

2. Forces of Friction
When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion
This is due to the interactions between the object and its environment
This resistance is called the force of friction

3. Forces of Friction, cont.
The force of static friction, ƒs, is generally greater than the force of kinetic friction, ƒk
The coefficient of friction (µ) depends on the surfaces in contact
Friction is proportional to the normal force
ƒs £ µs n and ƒk= µk n
These equations relate the magnitudes of the forces, they are not vector equations

4. Forces of Friction, final
The direction of the frictional force is opposite the direction of motion and parallel to the surfaces in contact
The coefficients of friction are nearly independent of the area of contact

5. Static Friction Static friction acts to keep the object from moving
If increases, so does
If decreases, so does
ƒs  µs n where the equality holds when the surfaces are on the verge of slipping
Called impending motion

6. Active Figure
AF_0501 static and kinetic frictional forces.swf

7. Kinetic Friction
The force of kinetic friction acts when the object is in motion
Although µk can vary with speed, we shall neglect any such variations
ƒk = µk n

8. Some Coefficients of Friction

9. Friction in Newton’s Laws Problems
Friction is a force, so it simply is included in the SF in Newton’s Laws
The rules of friction allow you to determine the direction and magnitude of the force of friction

10. Friction Example, 1
The block is sliding down the plane, so friction acts up the plane
This setup can be used to experimentally determine the coefficient of friction
µ = tan q
For µs, use the angle where the block just slips
For µk, use the angle where the block slides down at a constant speed

11. Friction Example 2
Image the ball moving downward and the cube sliding to the right
Both are accelerating from rest
There is a friction force between the cube and the surface

12. Friction Example 2, cont
Two objects, so two free body diagrams are needed
Apply Newton’s Laws to both objects
The tension is the same for both objects

13. Uniform Circular Motion
A force, , 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

14. 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

15. Active Figure
AF_0509 tangential velocity.swf

16. 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

17. 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

18. 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

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

20. 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

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

22. Non-Uniform Circular Motion
The acceleration and force have tangential components
produces the centripetal acceleration
produces the tangential acceleration

23. 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

24. 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

25. Active Figure
AF_0515 tangential and radial forces.swf

26. Motion with Resistive Forces
Motion can be through a medium
Either a liquid or a gas
The medium exerts a resistive force, , on an object moving through the medium
The magnitude of depends on the medium
The direction of is opposite the direction of motion of the object relative to the medium
nearly always increases with increasing speed

27. Motion with Resistive Forces, cont
The magnitude of can depend on the speed in complex ways
We will discuss only two
is proportional to v
Good approximation for slow motions or small objects
is proportional to v2
Good approximation for large objects

28. R Proportional To v The resistive force can be expressed as
b depends on the property of the medium, and on the shape and dimensions of the object
The negative sign indicates is in the opposite direction to

29. R Proportional To v, Example
Analyzing the motion results in

30. R Proportional To v, Example, cont
Initially, v = 0 and dv/dt = g
As t increases, R increases and a decreases
The acceleration approaches 0 when R ® mg
At this point, v approaches the terminal speed of the object

31. Terminal Speed To find the terminal speed, let a = 0
Solving the differential equation gives
t is the time constant and t = m/b

32. Active Figure
AF_0518 terminal speed.swf

33. R Proportional To v2
For objects moving at high speeds through air, the resistive force is approximately proportional to the square of the speed
R = 1/2 DrAv2
D is a dimensionless empirical quantity that is called the drag coefficient
r is the density of air
A is the cross-sectional area of the object
v is the speed of the object

34. R Proportional To v2, example
Analysis of an object falling through air accounting for air resistance

35. R Proportional To v2, Terminal Speed
The terminal speed will occur when the acceleration goes to zero
Solving the equation gives

36. Some Terminal Speeds

37. Fundamental Forces Gravitational force Electromagnetic forces
Between two objects
Electromagnetic forces
Between two charges
Nuclear force
Between subatomic particles
Weak forces
Arise in certain radioactive decay processes

38. Gravitational Force
Mutual force of attraction between any two objects in the Universe
Inherently the weakest of the fundamental forces
Described by Newton’s Law of Universal Gravitation

39. Electromagnetic Force
Binds atoms and electrons in ordinary matter
Most of the forces we have discussed are ultimately electromagnetic in nature
Magnitude is given by Coulomb’s Law

40. Nuclear Force
The force that binds the nucleons to form the nucleus of an atom
Attractive force
Extremely short range force
Negligible for r > ~10-14 m
For a typical nuclear separation, the nuclear force is about two orders of magnitude stronger than the electrostatic force

41. Weak Force Tends to produce instability in certain nuclei
Short-range force
About 1034 times stronger than gravitational force
About 103 times stronger than the electromagnetic force

42. Unifying the Fundamental Forces
Physicists have been searching for a simplification scheme that reduces the number of forces
1987 – Electromagnetic and weak forces were shown to be manifestations of one force, the electroweak force
The nuclear force is now interpreted as a secondary effect of the strong force acting between quarks

43. Drag Coefficients of Automobiles

44. Reducing Drag of Automobiles
Small frontal area
Smooth curves from the front
The streamline shape contributes to a low drag coefficient
Minimize as many irregularities in the surfaces as possible
Including the undercarriage