1. Concept of Force and Newton’s Laws of Motion 8.01 W02D2

2. Today’s Reading Assignment:
Young and Freedman Sections ,

3. Review: Relatively Inertial Reference Frames
Two reference frames with the
zero relative acceleration
One moving object has different
position vectors in different frames
Law of addition of velocities
Acceleration in either reference frame is the same
Spelling of “Inertial” corrected 3

4. Concept Question: Relatively Inertial Reference Frames
Suppose Frames 1 and 2 are relatively inertial reference frames.
An object that is at rest in Frame 2 is moving at a constant velocity in reference Frame 1.
An object that is accelerating in Frame 2 has the same acceleration in reference Frame 1.
An object that is moving at constant velocity in Frame 2 is accelerating in reference Frame 1.
An object that is accelerating in Frame 2 is moving at constant velocity in reference Frame 1.
Two of the above
None of the above
Grammar, punctuation corrected 4

5. C.Q. Answer: Newton’s First Law
Answer: (5). Both (1) and (2) are correct.
An object that is at rest in Frame 2 is moving at a constant velocity in reference Frame 1. The accelerations are the same in relatively inertial reference frames because the relative velocity of the two reference frames is constant hence the relative acceleration of the two reference frames is zero.
Grammar, punctuation corrected 5

6. Inertial Mass Inertial mass as a ‘quantity of matter’
Standard body with mass ms and SI units [kg]
Mass of all other bodies will be determined relative to the mass of our standard body.
Apply the same action to the standard body and an unknown body
Define the unknown mass in terms of ratio
Mass ratio is independent of the method used to produce the uniform accelerations.

7. Definition of Force Force is a vector quantity
The magnitude of the total force is defined to be
The direction of the vector sum of all the forces on a body is the same as the direction of the acceleration.
The SI units for force are newtons (N): 7

8. Superposition Principle
Apply two forces and on a body,
the total force is the vector sum of the two forces:
Notation: The force acting on body 1 due to the interaction between body 1 and body 2 is denoted by
Example: The total force exerted on body 3 due to the interactions with bodies 1 and 2 is: 8

9. Examples of Forces Gravitation Electric and magnetic forces
Elastic forces (Hooke’s Law)
Frictional forces: static and kinetic friction, fluid resistance
Contact forces: normal forces and static friction
Tension and compression 9

10. Newton’s First Law
Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
Newton’s First Law in relatively inertial reference frames:
If the vector sum of forces acting on an object at rest in Frame 2 is zero then the vector sum of forces acting on the object moving at constant speed in Frame 1 is also zero 10

11. Newton’s Second Law
The change of motion is proportional to the motive force impresses, and is made in the direction of the right line in which that force is impressed,
When multiple forces are acting,
In Cartesian coordinates: 11

12. Newton’s Third Law
To every action there is always opposed an equal reaction: or, the mutual action of two bodies upon each other are always equal, and directed to contrary parts.
Action-reaction pair of forces cannot act on same body; they act on different bodies. 12

13. Force Law: Newtonian Induction
Definition of force has no predictive content.
Need to measure the acceleration and the mass in order to define the force.
Force Law: Discover experimental relation between force exerted on object and change in properties of object.
Induction: Extend force law from finite measurements to all cases within some range creating a model.
Second Law can now be used to predict motion!
If prediction disagrees with measurement adjust model. 13

14. Empirical Force Law: Hooke’s Law
Consider a mass m attached to a spring
Stretch or compress spring by
different amounts produces
different accelerations
Hooke’s law:
Direction: restoring spring to equilibrium
Hooke’s law holds within some reasonable range of extension or compression 14

15. Table Problem: Hooke’s Law
Consider a spring with negligible mass that has an unstretched length 8.8 cm. A body with mass 150 g is suspended from one end of the spring. The other end (the upper end) of the spring is fixed. After a series of oscillations has died down, the new stretched length of the spring is 9.8 cm. Assume that the spring satisfies Hooke’s Law when stretched. What is the spring constant?

16. Force Law: Gravitational Force near the Surface of the Earth
Near the surface of the earth, the gravitational interaction between a body and the earth is mutually attractive and has a magnitude of
where is the gravitational mass of the body and g is a positive constant.
Do we want to discuss variations In g? 16

17. Force Laws: Contact Forces Between Surfaces
The contact force between two
surfaces is denoted by the vector
Normal Force: Component of the contact force perpendicular to surface and is denoted by
Friction Force: Component of the contact force tangent to the surface and is denoted by
Therefore the contact force can be modeled as a vector sum 17

18. Concept Question: Car-Earth Interaction
Consider a car at rest. We can conclude that the downward gravitational pull of Earth on the car and the upward contact force of Earth on it are equal and opposite because
the two forces form a third law interaction pair.
the net force on the car is zero.
neither of the above.
unsure

19. Concept Question: Normal Force
Consider a person standing in an elevator that is accelerating upward. The upward normal force N exerted by the elevator floor on the person is
larger than
identical to
smaller than
the downward force of gravity on the person.

20. Kinetic Friction
The kinetic frictional force fk is proportional to the normal force, but independent of surface area of contact and the velocity.
The magnitude of fk is
where µk is the coefficients of friction.
Direction of fk: opposes motion 20

21. Static Friction
Varies in direction and magnitude depending on applied forces:
Static friction is equal to it’s maximum value 21

22. Tension in a Rope
The tension in a rope at a distance x from one end of the rope is the magnitude of the action-reaction pair of forces acting at that point , 22

23. Table Problem: Tension in a Rope Between Trees
Suppose a rope is tied rather tightly between two trees that are separated by 30 m. You grab the middle of the rope and pull on it perpendicular to the line between the trees with as much force as you can. Assume this force is 1000 N, and the point where you are pulling on the rope is 1 m from the line joining the trees. What is the magnitude of the tension in the rope? 23

24. Concept of System: Reduction
Modeling complicated interaction of objects by isolated a subset (possible one object) of the objects as the system
Treat each object in the system as a point-like object
Identify all forces that act on that object

25. Free Body Diagram
Represent each force that is acting on the object by an arrow on a free body force diagram that indicates the direction of the force
Choose set of independent unit vectors and draw them on free body diagram.
Decompose each force in terms of vector components.
Add vector components to find vector decomposition of the total force

26. Newton’s Second Law: Strategy
Treat each object in the system as a point-like object
Identify all forces that act on that object, draw a free body diagram
Apply Newton’s Second Law to each body
Find relevant constraint equations
Solve system of equations for quantities of interest

27. Worked Example: Pulley and Inclined Plane
A block of mass m1, constrained to move along a plane inclined at angle ϕ to the horizontal, is connected via a massless inextensible rope that passes over a massless pulley to a bucket to which sand is slowly added. The coefficient of static friction is μs. Assume the gravitational constant is g. What is the mass of the bucket and sand (m2) just before the block slips upward?
The “phi” in the problem is not the same as the “phi” in the figure. We know that, but it won’t be clear to everyone. The subscript on “mu” is hard to read.

28. Solution: Pulley and Inclined Plane
Sketch and coordinate system
Free body force diagrams
To small to read easily. In upper figure, it would be best if the “+y” were “y_1”.

29. Newton’s Second Law Constraint Conditions Newton’s Second Law
Object 1:
Object 2:
Simplification:
Solution:

30. Concept Question: Varying Tension in String
A block of mass m1, constrained to move along a plane inclined at angle ϕ to the horizontal, is connected via a massless inextensible rope that passes over a massless pulley to a bucket to which sand is slowly added. The coefficient of static friction is μs. Assume the gravitational constant is g. What happens to the tension in the string just after the block begins to slip upward?
Increases
Decreases
Stays the same
oscillates
I don’t know
None of the above

31. Table Problem: Two Blocks and Two Pulleys
Two blocks 1 and 2 of mass m1 and m2 respectively are attached by a string wrapped around two pulleys as shown in the figure. Block 1 is accelerating to the right on a fricitonless surface. You may assume that the string is massless and inextensible and that the pulleys are massless. Find the accelerations of the blocks and the tension in the string connecting the blocks.
Should this be a table problem

32. Lecture Demo: Block with Pulley and Weight on Incline B24

33. Table Problem: Painter on Platform
A painter pulls on each rope with a constant force F. The painter has mass m1 and the platform has mass m2.
Draw free body force diagrams for the platform and painter. Is there something wrong with this picture?
Find the acceleration of the platform.

34. Source: why you need to know physics

35. Next Reading Assignment: W02D3
Young and Freedman (Review)

36. Appendix: Newton’s Second Law Detailed Problem Solving Strategy

37. Methodology for Newton’s 2nd Law
Understand – get a conceptual grasp of the problem
Sketch the system at some time when the system is in motion.
Draw free body diagrams for each body or composite bodies:
Each force is represented by an arrow indicating the direction of the force
Choose an appropriate symbol for the force

38. II. Devise a Plan Choose a coordinate system:
Identify the position function of all objects and unit vectors.
Include the set of unit vectors on free body force diagram.
Apply vector decomposition to each force in the free body diagram:
Apply superposition principle to find total force in each direction:
Minor changes in MathType formulas.

39. II. Devise a Plan: Equations of Motion
Application of Newton’s Second Law
This is a vector equality; the two sides are equal in magnitude and direction.
Minor MathType changes

40. II. Devise a Plan (cont’d)
Analyze whether you can solve the system of equations
Common problems and missing conditions.
Constraint conditions between the components of the acceleration.
Action-reaction pairs.
Different bodies are not distinguished.
Design a strategy for solving the system of equations.
I don’t understand the first item, or how to fix it.

41. III. Carry Out your Plan Hints:
Use all your equations. Avoid thinking that one equation alone will contain your answer!
Solve your equations for the components of the individual forces.

42. IV. Look Back Check your algebra Substitute in numbers
Check your result
Think about the result: Solved problems become models for thinking about new problems.