1. Physics 121, Sections 9, 10, 11, and 12 Lecture 3
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2. Physics 121, Sections 9, 10, 11, and 12 Lecture 2
Today’s Topics:
Homework 1: Due Friday Sept. 6 @ 6:00PM
Ch.1: # 4, 10, 14, 19, 31, and 33.
Ch.2: # 1, 23, 37, and 63.
Chapter 2: Forces and vectors
Types of forces
Newton’s Laws of motion
Net force and vector addition
Contact force and tension
Chapter 3: Forces and motion along a line
Position, velocity, acceleration 1

3. Problem Solution Method:
Five Steps:
Focus the Problem
- draw a picture – what are we asking for?
Describe the physics
what physics ideas are applicable
what are the relevant variables known and unknown
Plan the solution
what are the relevant physics equations
Execute the plan
solve in terms of variables
solve in terms of numbers
Evaluate the answer
are the dimensions and units correct?
do the numbers make sense?

4. Chap.2: Forces and vectors
In classical mechanics
Need to study interactions between objects
Described by forces
We have an idea of what a force is from everyday life.
Physicist must be precise.
A force is that which causes a body to accelerate.
(See Newton’s Second Law)
A Force is a push or a pull.
A Force has magnitude & direction (vector). F

5. Fundamental Forces Example of Forces Fundamental Forces
Hooke’s law for ideal spring: F = -k x
Units of a force are 1 N= 1 kg m/s2
Fundamental Forces
Gravity (more later)
For motion of planets, etc.
Strong and weak nuclear forces (not here !)
Explains behavior of nucleus in atoms
Electromagnetic force (next semester in PHY122)
Relevant for electric systems, chemical properties, etc.

6. The Laws of Motion
Isaac Newton ( ) published Principia Mathematica in In this work, he proposed three “laws” of motion:
Law 1: An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame.
Law 2: For any object, FNET = F = ma
Law 3: Forces occur in pairs: FA ,B = - FB ,A
(For every action there is an equal and opposite reaction.)
More in following chapters

7. Net Force: adding vectors
A Force has magnitude & direction (vector).
Adding forces is like adding vectors (more next chapter)
The net force is obtained by adding all forces
Adding collinear vectors
2 vectors in the same direction
Magnitude is the sum of both magnitudes
Direction remains the same
2 vectors in opposite direction
Magnitude is the absolute value of the difference of both magnitudes
Direction is the same as the longest vector
Sum of 2 vectors is zero if they have opposite directions and same magnitude
500 N
200 N
300 N
100 N
200 N
300 N
500 N

8. The Free Body Diagram Newton’s 2nd Law says that for an object F = ma.
Key phrase here is for an object.
So before we can apply F = ma to any given object we isolate the forces acting on this object:
We obtain the FBD

9. FBD: an example A mass is suspended to the ceiling with a ropemg T m
A mass is suspended to the ceiling with a rope
The FBD of the mass is simply given by all forces on acting on it mg T m

10. Internal and External Forces
Consider a system
E.g. a baseball
All atoms/particles inside interact with each other
Atom 1 acts on atom 2 with F21
But atom 2 also acts on atom 1 with F12
Newton’s 3rd law says that F12 = - F21
So the net force is zero … same for all pairs of particles
All interanl forces add up to zero
Only external forces remains
E.g., gravity or the contact of a stick !
We will deal with external forces mostly

11. Force … We will consider two kinds of forces
Field Forces (Non-Contact): (action through “empty” space)
Moon and Earth
Gravity
Electricity
Contact force: (physical contact between objects)
This is the most familiar kind.
Kicking a ball
I push on the desk.
The ground pushes on the chair...
On a microscopic level, all forces are non-contact

12. Action at a distance
Gravity:
next

13. Newton’s Law of Universal Gravitation
Every particle in the Universe attracts every other particle with a force that is directly proportional the product of their masses and inversely proportional to the square of the distance between them.
Constant of Universal gravitation:
F12
F21 m2
r12 m1

14. Weight W Near the surface of the Earth r ≈ RE (Earth radius)
So the gravitation force between an object of mass m and the Earth is simply
With the numbers for Earth:

15. Contact forces: Objects in contact exert forces.
Convention: Fa,b means “the force acting on a due to b”.
So Fhead,thumb means “the force on the head due to the thumb”.
Fhead,thumb
next

16. Examples of Contact Forces
Normal forces
Forces due to a surface pushing on an object
perpendicular to the surface
Box on a table
Box on inclined slope
FBD FX
N = mg mg FX
N = mg mg mg N
mg sin 
mg cos   m a 
next

17. Examples of Contact Forces
Friction
What does it do?
It opposes motion!
How do we characterize this in terms we have learned?
Friction results in a force in a direction opposite to the direction of motion! ma
FAPPLIED
fFRICTION mg N i j

18. Friction...
Friction is caused by the “microscopic” interactions between the two surfaces:

19. Friction... Force of friction acts to oppose motion:
Parallel to surface.
Perpendicular to Normal force. j N F i ma fF mg

20. Model for Sliding Friction
The direction of the frictional force vector is perpendicular to the normal force vector N.
The magnitude of the frictional force vector |fF| is proportional to the magnitude of the normal force |N |.
|fF| = K | N | ( = K|mg | in the previous example)
The “heavier” something is, the greater the friction will be...makes sense!
The constant K is called the “coefficient of kinetic friction”.

21. Tools: Ropes & Strings Can be used to pull from a distance.
Tension (T) at a certain position in a rope is the magnitude of the force acting across a cross-section of the rope at that position.
The force you would feel if you cut the rope and grabbed the ends.
An action-reaction pair
(more later about 3rd law) T
cut T T

22. Tools: Pegs & Pulleys Used to change the direction of forces.
An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude: F1
ideal peg
or pulley
| F1 | = | F2 | F2

23. Chapter 3: Position / Displacement
Displacement is just change in position.
X = xf - xi
Displacement is a vector quantity
A vector quantity has both magnitude and direction
A scalar quantity has only magnitude and no direction
10 meters
15 meters
Joe xf xi O
x = 5 meters { 2

24. Average Velocity
The average velocity v during the time interval t is defined as the displacement x divided by t.

25. Average Velocity
xi = 20 m, xf = 60 m, and t = 5.0 s

26. Average Speed
In every day usage, speed and velocity are interchangeable.
In Physics we have a clear distinction between speed and velocity.
The average speed is a scalar, the average velocity is a vector

27. Instantaneous Velocity
The instantaneous velocity v is defined as the limit of the average velocity as the time interval t becomes infinitesimally short

28. v = |v| Instantaneous Speed
The instantaneous speed of an object, which is a scalar quantity, is defined as the magnitude of the instantaneous velocity
v = |v|

29. Recap: Average speed and velocity
Average velocity = total distance covered per total time,
Speed is just the magnitude of velocity.
The “how fast” without the direction.
Instantaneous velocity, velocity at a given instant
Active Figure 1

30. Lecture 3, ACT 1 Average Velocity
x (meters) 6 4 2
t (seconds) -2 1 2 3 4
What is the average velocity over the first 4 seconds ?
A) -2 m/s
B) 4 m/s
C) 1 m/s
D) not enough
information to
decide.

31. Lecture 3, ACT 2 Instantaneous Velocity
x (meters) 6 4 2 -2
t (seconds) 1 2 3 4
What is the instantaneous velocity at the fourth second ?
A) 4 m/s
B) 0 m/s
C) 1 m/s
D) not enough
information to
decide.

32. Average Acceleration
The average acceleration a during the time interval t, is defined as the change in velocity v divided by the t
It is a vector …

33. Instantaneous Acceleration
The instantaneous acceleration a is defined as the limit of the average acceleration as the time interval t goes to zero

34. Kinematic Variables Measured with respect to a reference frame.
(x-y axis)
Measured using coordinates (having units).
Many kinematic variables are vectors, which means they have a direction as well as a magnitude.
Vectors denoted by boldface v or arrow

35. Motion in 1 dimension In 1-D, we usually write position as x .
Since it’s in 1-D, all we need to indicate direction is + or .
Displacement in a time t = tf - ti is x = xf - xi t x ti tf
x
t xi xf
some particle’s trajectory in 1-D

36. 1-D kinematics Velocity v is the “rate of change of position”
Average velocity vav in the time t = tf - ti is: t x t1 t2
x x1 x2
trajectory
t
Vav = slope of line connecting x1 and x2.

37. 1-D kinematics... Consider limit tf - ti  0
Instantaneous velocity v is defined as: t x t1 t2
x x1 x2
t
so v(t2 ) = slope of line tangent to path at t2.

38. 1-D kinematics...  Acceleration a is the “rate of change of velocity”
Average acceleration aav in the time t = t2 - t1 is:
If a = aav is constant, and set t1=0, v1= v0 , and then v2= v 

39. Average Velocity Velocity is: v = v0 + at (for constant acceleration)
vav v0

40. 1-D kinematics We saw that v = x /  t
After a little algebra we have:
x = v   t
Graphically, we can add up lots of small rectangles:
x(t) t +
+...+
= displacement

41. 1-D kinematics vav = x /  t For constant acceleration a we find:
Recall :
Again, set t1=0 and t2=t so that  t = t
And so

42. Recap
So for constant acceleration we find: x v t a t

43. Recap of today’s lecture
Finished Chapter 2: Forces and vectors
Types of forces
Newton’s Laws of motion
Net force and vector addition
Contact force and tension
Started Chapter 3:
Position, velocity, acceleration
Homework 1 on WebAssign 27