Tuesday June 7, 2005 1 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt PHYS 1443 – Section 001...

Tuesday June 7, 2005 1 PHYS 1443-001, Summer I 2005Dr. Andrew Brandt

PHYS 1443 – Section 001Lecture #5

Tuesday June 67 2005Dr. Andrew Brandt

• Reference Frames• Short Review• Newton’s Laws/Force


Announcements• Homework+tests:

– First homework attempted by 11/12, most people did well

– Second HW on ch3 due today 5pm (ch4 HW due 6/13 at 2pm, but start it early)

– First test on Ch 1-3 Weds June 8 at 8:30 am


Maximum Range and Height• What are the conditions that give maximum height and

range of a projectile motion?

g

vh ii

2

sin 22 This formula tells us that the maximum height can be achieved when i=90o!!!

g

vR ii 2sin2

This formula tells us that the maximum range can be achieved when 2i=90o, i.e., i=45o!!!


Observations in Different Reference FramesResults of physical measurements in different reference frames could be differentObservations of the same motion in a stationary frame would be different than the ones made in the frame moving together with the moving object.Consider that you are driving a car. To you, the objects in the car do not move while to the person outside the car they are moving with the same speed and direction as your car.

O

Frame S

r’

O’

Frame S’The position vector r’ is still r’ in the moving frame S’.no matter how much time has passed!!

v0

The position vector r is no longer r in the stationary frame S when time t has passed. tvrtr 0')(

v0t

rHow are these position vectors related to each other?


Relative Velocity and AccelerationThe velocity and acceleration in two different frames of references can be denoted, using the formula in the previous slide:

O

Frame S

r’

O’

Frame S’v0

v0t

r

tvrr 0' Galilean transformation equation

What does this tell you?

The acceleration measured in two frames are the same when the frames move at a constant velocity with respect to each other!!!

The earth’s gravitational acceleration is the same in a frame moving at a constant velocity wrt the earth.

0'

vdt

rd

dt

rd

0' vvv

dt

vd

dt

vd

dt

vd 0'

constant is when ,' 0vaa


Review: Ch 1-2

tavtv xxixf

2

2

1tatvxx xxiif

1

2f xi xf xix x v t v v t

2 2 2f f ix xi xv v a x x

Velocity as a function of time

Displacement as a function of velocity and time

Displacement as a function of time, velocity, and acceleration

Velocity as a function of displacement and acceleration

Significant figures, displacement, velocity, acceleration,1-D motion, free fall, constant acceleration equations:


Chapter 3

g

vR ii 2sin2

Coordinate systems, vectors (defn., operations, componentsunit vectors), 2-D motion, projectile motion, reference frames

Range+Max height

2 2sin

2i iv

hg


Force (Ch 4)We’ve been learning kinematics; describing motion without understanding the cause of the motion. Now we are going to learn dynamics!!

What is FORCE?

FORCEs are what cause an object to move

FORCEs cause a change in the velocity of an object!!

The above statement is not entirely correct. Why?Because when an object is moving with a constant velocity no force is exerted on the object!!!

What does this statement mean?

When there is force, there is change of velocity. Forces cause acceleration.

What happens there are several forces being exerted on an object?

Forces are vector quantities, so vector sum of all forces, the NET FORCE, determines the motion of the object.F1 F2 NET FORCE,

F= F1+F2

When net force on an object is 0, it has constant velocity and is at its equilibrium!!


More ForceThere are various classes of forces

Contact Forces: Forces exerted by physical contact with objectsExamples of Contact Forces: Baseball hit by a bat, Car collisionsField Forces: Forces exerted without physical contact of objectsExamples of Field Forces: Gravitational Force, Electromagnetic forceWhat are possible ways to measure strength of Force?A calibrated spring whose length changes linearly with the force exerted .Forces are vector quantities, so addition of multiple forces must be done following the rules of vector addition.


Newton’s First Law and Inertial FramesAristotle (384-322BC): A natural state of a body is rest. Thus force is required to move an object. To move faster, ones needs higher force.Galileo’s statement on natural states of matter: Any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of retardation are removed!!Galileo’s statement is formulated by Newton into the 1st law of motion (Law of

Inertia): In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity.

A frame of reference that is moving at constant velocity is called an Inertial Frame

What does this statement tell us? • When no force is exerted on an object, the acceleration of the object is 0. • Any isolated object, an object that does not interact with its surrounding, is

either at rest or moving at a constant velocity.• An object maintains its current state of motion, as long as there is no force that

interferes with the motion. This tendency is called the Inertia.


MassMass: A quantity of matter, or, more usefully, a

measure of the inertia of a body 1. Independent of the object’s surroundings: The same no “matter” where you go.2. Independent of the method of measurement:

1

2

2

1

a

a

m

mThe same forces applied to two

different masses result in different acceleration depending on the mass.

The heavier (more massive) an object, the bigger the inertia.It is harder to change the motion of a heavier object than a lighter one.

Note that mass and weight of an object are two different quantities!Weight of an object is the magnitude of gravitational

force exerted on the object. Not an inherent property of an object.Weight will change if you measure on the Earth or on the moon.


Newton’s Second Law of Motion

amFi

i

The acceleration of an object is directly proportional to the net force exerted on it and is inversely proportional to the object’s mass. How do we write the above statement in a mathematical expression?

xi

ix maF Since it’s a vector expression, each component should also satisfy it:From the above vector expression, what do you conclude the dimension and unit of force are?

]][[ am

]][[]][[][ 2LTMamForce

The dimension of force is The unit of force in SI is For simplicity, we

define a new derived unit called, a Newton (N)

yi

iy maF zi

iz maF

lbssmkgN4

1/11 2

]][[ 2LTM2/ smkg


Acceleration

Example of Newton’s 2nd Law of MotionWhat constant net force is required to bring a 1500kg car to rest from a speed of 100km/h within a distance of 55m?

This is a one dimensional motion. Which kinetic formula do we use to find acceleration?

What is given?

Thus, the force needed to stop the car is

xF

hkmvxi /100

ifxxixf xxavv 222

What do we need to know to figure out the force? Acceleration!!

Initial speed: sm /28

Displacement: mxxx if 55

Final speed: smvxf /0

if

xixfx xx

vva

2

22

22

/1.7552

/28sm

m

sm

Nsmkgmax42 101.1/1.71500

How are stopping distance and force related?

x

xixfif a

vvxxx

2

22

x

xixf

ma

vvm

2

22 2

2xi

x

m v

F

•Linearly proportional to the mass of the car•Squarely proportional to the speed of the car•Inversely proportional to the force by the brake


Acceleration Vector a

F

F

Example for Newton’s 2nd Law of MotionDetermine the magnitude and direction of acceleration of a hockey puck with a mass of 0.30kg, being pulled by two forces, F1 and F2, with magnitudes of 8.0 N and 5.0 N, respectively, as shown in the picture.

xF1

xF

Components of F1

Components of F2

Components of total force F

xa Magnitude and direction of acceleration a

yF1

xF2

yF2

yF

ya a r

a r

1F cos r

1F sin r

2 2F cos r

2 2F sin r

1 2x xF F xma

1 2y yF F yma

xF

m 28.7

29 /0.3

m s yF

m 25.2

17 /0.3

m s 22

x ya a 2 229 17

234 /m s1tan y

x

a

a

1 17tan 30

29

o

x ya i a j

229 17 /i j m s

8.0 cos 60 4.0N

8.0 sin 60 6.9N

5.0 cos 20 4.7N

5.0 sin 20 1.7N

4.0 4.7 8.7N 6.9 1.7 5.2N


Gravitational Force and Weight

Since weight depends on the magnitude of gravitational acceleration, g, it varies depending on geographical location.

The attractive force exerted on an object by the Earth

Gravitational Force, Fg

amF G

Weight of an object with mass M is W

By measuring the forces one can determine masses.

gm

FG r

gM r

Mg


Some Basic Information

Normal Force, n:

When Newton’s laws are applied, external forces only are of interest!!

Why? Because, as described in Newton’s first law, an object will keep its current motion unless a net non-zero external force is applied.

Tension, T:

Reaction force that reacts to gravitational force due to the surface structure of an object. Its direction is perpendicular to the surface.The reactionary force by a stringy object against an external force exerted on it.

A graphical tool which is a diagram diagram of external forces on an objectof external forces on an object and is extremely useful in analyzing forces and motion!! Drawn only on an object.

Free-body diagram

assume masslesstaut strings


Free Body Diagrams and Solving Problems• Free-body diagram: A diagram of vector forces acting on an object A great tool to solve a problem using forces or using dynamics1. Select a point on an object in the problem2. Identify all the forces acting only on the selected object3. Define a reference frame with positive and negative axes specified4. Draw arrows to represent the force vectors on the selected point5. Write down the net force vector equation6. Write down the forces in components to solve the problem(s) No matter for which object we choose to draw the diagram, the results should be the same, as

long as they are from the same motion

MWhich one would you like to select to draw the FBD?What do you think are the forces acting on this object?

Gravitational forcegMF G A force supporting the object exerted by the floor

Me

Which one would you like to select to draw theFBD?What do you think are the forces acting on this elevator?

NF

gMF G

Gravitational force The force pulling the elevator (Tension)

mWhat about the box in the elevator? Gravitational

forceNormal force

NF

TF

gMF G gmF GB

TF

gMF G

NF

gmF BG

NF

Tuesday June 7, 2005 1 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt PHYS 1443 – Section 001...

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