## I. 3.1: MASS, FORCE, AND NEWTON’S LAWS: study of dynamics

### A. Force

Push or pull exerted by one object on another (i.e. tension, friction, gravitational electrostatic, air resistance)

### B. Newton’s laws

1. 1: An object at rest stays at rest, an object in motion stays in motion (if a = 0, v is constant; no net force, no acceleration)

a) Law of inertia

b) Mass of an object is the quantitative measurement of its inertia

2. 2: *F = m*a*

^{2}= N]

a) F refers to net force, or the sum of all forces

3. 3: With every force, there is an equal and opposite force

a) If object 1 exerts a force, F1-on-2, on object 2, object 2 will exert a force, F2-on-1, on object 1

b) These forces are equal but in opposite directions; are called an action-reaction pair

### C. Things to remember about Newton’s laws:

1. 1st law vs 3rd law: 1st law implies forces are on a single object; 3rd law implies each force must be on a different object (on the objects of the action-reaction pair)

2. 3rd law: if 2 forces are equal and opposite, it does not automatically mean they are an action-reaction pair; these forces have to act on each other in order to be this

3. 3rd law: although the 2 forces must be equal, their effects are not necessarily equal (remember, if the mass of one is different, its acceleration will be different too)

## II. 3.2: NEWTON’S LAW OF GRAVITATION

### A. Mass and weight are not interchangeable; weight is a force (gravitational force exerted on mass)

### B. w = mg (weight = mass*gravitational acceleration), weight is in newtons (N)

### C. Newton’s law of gravitation:

1. Every object exerts gravitational pull on every other object

a) Is proportional to the product of the object’s masses

b) Inverse-square law: inversely proportional to the square of the distance between them (between the centers of the objects)

c) Constant of proportionality is G (universal gravitational constant)

d) **F _{grav} = G(Mm)/r^{2}** (r = distance between centers)

e) Combine F_{grav} = G(Mm)/r^{2} with w = mg (w = F_{grav}):

(1) mg = G(Mm)/r^{2}

(2) g = GM/r^{2} → the value of gravitational acceleration on earth = 10 m/s^{2}

## III. 3.3: FRICTION

### A. Normal force (N, FN)

The opposite force exerted against an object exerting a force (usually gravitational); EX – book on table

1. Action-reaction pairs:

a) Reaction force to F_{table-on-book} is F_{book-on-table}

b) Reaction force to to F_{earth-on-book} is F_{book-on-earth}

c) Reaction force to F_{table-on-book} *is not* F_{earth-on-book}

2. Remember: action-reaction pairs always act on different objects!!

3. Normal force is the perpendicular component of the contact force exerted by a surface on an object

4. F_{N} = m*a (usually mass*gravitational acceleration)

### B. 2 types of friction

1. **Kinetic (sliding) friction** – the mechanical friction between 2 surfaces due to roughness

a) **Coefficient of kinetic friction (μk)** – depends on what the surfaces are made of, experimentally determined, no units

b) **F _{f} = μ_{k}F_{N} ** → force of kinetic friction, not vector! Magnitude only

(1) The direction of kinetic friction is always parallel to surface and opposite of velocity

2. **Static friction** – the attraction between electrical atoms of 1 surface with those of another

a) Maximum coefficient of static friction (μs) → on MCAT, this is always > than μk!

(1) Starting to push something is always harder than continuing to push it

b) **F _{f, max} = μ_{s}F_{N} ** → this is the max, because any force less than max will do nothing

(1) If F_{f, max} = 200 N, and you exert 100 N, the static friction will only be 100 N

## IV. 3.4: INCLINED PLANE

### A. FN = mgcosθ

### B. w = mg

### C. F ǁ to plane = mgsinθ

### D. EX:

Block mass m = 4 kg is placed at the top of a frictionless ramp of incline angle 30º and length 10m.

1. What is the block’s acceleration down the ramp?

a) F = m*a; a = F/m

b) a = mgsinθ/m = gsinθ = (10 m/s^{2})(0.5) = 5 m/s^{2}

2. How long will it take for the block to slide to the bottom?

a) T = ?; a = 5; d = 10 v_{0} = 0

b) d = v_{0}t + ½at^{2}

c) 10 = 0 + ½(5)t^{2}

d) 10 = 2.5t^{2}

e) 4 = t2; t = 2 s

## V. 3.5: PULLEYS

### A. Pulleys change direction of tension (FT) of the rope that pulls on the object that the rope attaches to

### B. Pulleys can also decrease the force necessary to lift an object (Σ↑FT = m*g)

### C. EX:

Multiple pulley system

1. To pull the plank up at constant speed, how much force is needed?

2. If v is constant, a=0, therefore net force = 0

3. Therefore, Σ FT↑ = M*a

4. FT↑ = FT↓

5. 6FT = M*g

### D. EX:

1 pulley, 2 masses: if m = 5kg and M = 10 kg what is acceleration when released from rest?

1. Draw a force diagram

2. Choose direction to call (+); usually which direction the objects will move

3. Find F_{NET} and set it equal to m*a

4. Figure out the m*a for each

5. F_{NET(m)} = m*a = F_{T} – m*g

6. F_{NET(m)} = M*a = M*g – F_{T}

7. Add equations together:

a) M*g – m*g = M*a + m*a

b) g(M – m) = a (M + m)

c) a = G(M – m)/(M + m)

d) a = 5g/15 = g/3

## VI. 3.6: SUMMARY OF FORMULAS:

**F**_{NET}= m*a**w = m*g****F**→ w = F_{grav}= G*Mm/r^{2}_{grav}= m*g → m*g = G*Mm/r^{2}→**g = GM/r**^{2}**F**_{f}= μ_{k}F_{N}**F**(μ_{f, max}= μ_{s}F_{N }_{s, max}> μ_{k})**F**_{grav, ‖ to inclined plane}= mg*sinθ