– Concepts of Mass and Force – Newton’s Three Laws

Chapter 5. Force and Motion I

• Newton’s Laws

– Concepts of Mass and Force – Newton’s Three Laws

But first, let’s review the last lecture..

Summary of the last lecture

1. Projectile Motion

2. Circular motion

3. Relative Motion

• x = x₀ + v₀t

• v = v_0x

• y = y₀ + v_0yt - 1/2 gt²

• v_y = v_0y^-gt

• v_y² = v_0y² - 2g Δy

PB PA BA PB PA BA

r = r − r → v = v − v

v = rω G JG

v ω = r

JG G

ω=2πf [S^-1] Period T

1

2 2

T f

r v π ωπ

=

=

=

2 2

d v d

a v v

dt dt a a v r v

r θ ω

ω ω

= = =

= = = =

G G

G JG

G

Isaac Newton (1642 ~ 1727)

Question

You are driving a car up a hill with constant velocity. On a piece of paper, How many forces are acting on the car?

1 2 3 4 5

W

f F_N

V

correct

Question

The net force on the car is,

1. Zero

2. Pointing up the hill 3. Pointing down the hill

4. Pointing vertically downward 5. Pointing vertically upward

W

f

F_{N V}

W f

F_N ΣF = ma = 0

correct

Question

You are driving a car up a hill with constant acceleration.

How many forces are acting on the car?

1 2 3 4 5

W

f F_N

a correct

2) Compare the magnitudes of the acceleration you experience, a_A, to the magnitude of the acceleration of the spacecraft, a_S, while you are

pushing:

1. a_A = a_S 2. a_A > a_S 3. a_A < a_S

Question

Suppose you are an astronaut in outer space giving a brief push to a spacecraft whose mass is bigger than your own .

1) Compare the magnitude of the force you exert on the spacecraft, F_S, to the magnitude of the force exerted by the spacecraft on you, F_A, while you are pushing:

1. F_A = F_S 2. F_A > F_S 3. F_A < F_S

correct

correct a = F/m

F same ⇒ lower mass give larger a Third Law!

Second Law!

Newton’s First Law

R The motion of an object does not change unless it is acted upon by a net force.

• If v=0, it remains 0.

• If v is some value, it stays at that value.

R Another way to say the same thing:

• No net force (F_net = 0) ⇔

• velocity is constant.

• acceleration is zero.

Mass or Inertia

R Mass (m) is the property of an object that measures how hard it is to change its motion.

R Units: [M] = kg

Examples of 1st Law

• braking car vs. train

• car vs. bus going around curve

• push on light or heavy object

Newton’s Second Law

F

a = m G JG

F

= ∑ F ma =

JG JG G

Cause (원인)

Result (결과)

Question

An airplane is flying from Seoul airport to Jegu. Many forces act on the plane, including weight (gravity), drag (air resistance), the trust of the engine, and the lift of the wings. At some point during its trip the velocity of the

plane is measured to be constant (which means its

altitude is also constant). At this time, the totaltotal force on the plane:

1. is pointing upward 2. is pointing downward 3. is pointing forward 4. is pointing backward 5. is zero

lift

weight

drag thrust

correct

gravity would be working against the airplane flying at a level altitude, so the total force would need to be upward.

Gravity never goes away, so the force on the plane will be pointing downwards.

The force has to be pointing forward since the velocity is moving forward.

the up and down forces are zero because the plane is not going up or down, and the plane is pushing back against the air propelling it forward.

something to think about…

the plane is constantly pointing down because the earth is round and constant altitude would mean it has to decrease its altitude constantly to keep a constant distance from the ground

True, but the effect is small.. On a cross-country trip (300 miles), a “flat” trip would have distance to center of earth varying by 0.5%

Questions

(common misconceptions)

Example 1

F₁ M M=10 kg, F₁=200 N

Find a.

a = F_net/M = 200N/10kg = 20 m/s²

M=10 kg

F₁=200 N F₂ = 100 N Find a.

F₁ M F₂

a = F_net/M = (200N-100N)/10kg = 10 m/s²

Example 2

• A force F acting on a mass m₁ results in an acceleration a₁. The same force acting on a different mass m₂ results in an acceleration a₂ = 2a₁. What is the mass m₂?

(a)(a) 2m₁ (b) m(b) ₁ (c)(c) 1/2 m₁

F m₁ a₁

F a₂ = 2a₁

m₂

• F=ma

• F= m₁a₁ = m₂a₂ = m₂(2a₁)

• Therefore, m₂ = m₁/2

• Or, in words… twice the acceleration means half the mass.

Examples of Force

1. Gravity (중력, 혹은 무게)

FG12

R mg m F GM

E

E ≡

= ₂

~ Gravitational Force

m₁ m₂

FG21

21

12 F

FG G

−

= : Gravitational force between massive objects

2 12 12

1 2

12 rˆ

r m FG = Gm

2 21 21

2 1

21 rˆ

r m

FG = Gm rˆ¹² = − rˆ²¹

On the Earth

R Earth

m M_E = 6 ×10²⁴ kg

R_E = 6.4×10⁶ m

( ) ²

24 11

8 10 9

6 10

67

6 . . m / sec

g = GM ^E = × ⁻ ⋅ × =

G = universal gravitation constant = 6.67 x 10^-11 N-m²/kg²

Examples of Force

2. Normal force (수직힘, 수직항력, 법선력)

book at rest on table:

What are forces on book?

W

• Weight is downward

• System is “in equilibrium” (acceleration = 0 ⇒ net force = 0)

• Therefore, weight balanced by another force F_N

• F_N = “normal force” = force exerted by surface on object

• F_N is always perpendicular to surface and outward

• For this example F_N = W

Another Example:

• Book at rest on table:

• Push down with F_hand

• What is F_N? W

F_hand

F_N

• Equilibrium:

• F_N + W + F_hand = 0

• ⇒ F_N = W+F_hand What is direction of F_N?

Examples of Force

3. Friction (쓸림, 쓸림힘, 마찰력)

(접촉마찰력)

Examples of Force

4. Tension (장력)

M M : T = Ma

H : T = F H F

M

m M

F

M : T = Ma

m : T – mg = -ma

Newton’s Third Law

R For every action, there is an equal and opposite reaction.

• Finger pushes on box

• F_finger→box = force exerted on box by finger F_finger→box

F_box→finger

• Box pushes on finger

• F_box→finger = force exerted on finger by box

• Third Law:

F_box→finger = - F_finger→box

(두 물체가 상호작용할 때서로에게 작용하는 힘(짝힘)은 항상 크기가 같고 방향이 반대이다.)

Newton's Third Law...

• FF_{A ,B} = - FF_{B ,A}. is true for all types of forces

FF_w,m FF_m,w

FF_f,m FF_m,f

Example of Bad Thinking

• Since FF_m,b = - FF_b,m why isn’t FF_net = 0, and aa = 0 ?

a ??a ??

FF_b,m FF_m,b

ice

Example of Good Thinking

• Consider only the box!only the box –– FF_{on box} = maa_box= FF_m,b

– Free Body Diagram (which means a diagram including all forces existing) What about forces on man?

aa_boxbox

FF_b,m FF_m,b

ice

(a) 참외에 작용하는 힘은?

(b) 참외와 지구 사이에 작용하는 제3법칙의 짝힘은?

F_CE (중력), F_CT (식탁이 주는 수직힘)

이 두 힘은 짝힘인가? 아니다!

F_CE (지구 인력) = - F_EC (참외 인력)

(c) 참외와 식탁 사이에 작용하는 제3법칙의 짝힘은?

F_TC (참외가 식탁에 가한 수직힘) = - F_CT (식탁이 참외에 가한 수직힘)

Newton's Third Law...

Summary

• Newton’s First Law:

The motion of an object does not change unless it is acted on by a net force

• Newton’s Second Law:

F_net = ma

• Newton’s Third Law:

F_a,b = - F_b,a

• Types of Forces – Gravity

– Normal Force – Friction

M₁ = 2 kg

Find acceleration of blocks and tension T₁

Answers: a = 4 m/s² and T₁= 16 N M₂ = 4 kg

F = 24 N T₁

Application of Newton's Laws

Example

M₁ = 4 kg

Turn blocks around….

Find acceleration of blocks and tension T₁ Answers: a= 4 m/s² (the same)

and T₁= 8 N (smaller) M₂ = 2 kg

T=24 N T₁

Example

Find acceleration and normal force

Answers: a= g sin(θ) F_N = mg cos (θ) θ

m a

mg

Example : Pulley Problem I

What is the tension in the string?

A) T<W B) T=W

C) W<T<2W D) T=2W

E) T>2W

W W

Pull with force = W W

Same answer T

Example : Pulley Problem II

What is the tension in the string?

A) T<W B) T=W

C) W<T<2W D) T=2W

E) T>2W

T

T

Example : Pulley Problem III

• Identify forces

• Set up axes

• Write F _net = ma for each axis

• Solve!

• and, Verify

Procedure for Solving Problems

Gravity Tension

Normal force Friction

M

m

Æ 예) 차원 비교? T < mg ?

m₂

m₁ m₁g

T

m₂g T

a m g

m T

F₁ = − ₁ = ₁ a m T

g m

F₂ = ₂ − = ₂

+) (

)

(

)

m g m

m a m

2 1

1 2

+

= − , ^g

m m

m T m

2 1

2

2 1

= +

m₁

m₂

F P P

a m P

F₂ = = ₂ a m P

F

F₁ = − = ₁

+)

(

)

2

1 m

m a F

= + , ^P _m ^m _m ^F

2 1

2

= + Atwood’s Machine

m (1) a

T

mg

T

mg (2) a

(1) F = T - mg = ma T = m(g + a)

(2) F = mg - T = ma T = m(g - a)

Example : Weighting in an Elevator