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PHYSICS 231
INTRODUCTORY PHYSICS I
Lecture 4
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• Scalars vs. Vectors• Vectors A: (Ax, Ay) or |A| & θ
• Addition/Subtraction
• Projectile Motion• X-direction: ax = 0 (vx = constant)• Y-direction: ay=-g• Parabolic trajectory
Main points of last lecture
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Example 3.4a
h
Dθ
v0
The X-component of v is smallest (in magnitude) at:a) Ab) Bc) Cd) Same at all points
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Example 3.4b
h
Dθ
v0
The Y-component of v is smallest (in magnitude) at:a) Ab) Bc) Cd) Same at all points
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Example 3.4c
h
Dθ
v0
The acceleration is smallest (in magnitude) at:a) Ab) Bc) Cd) Same at all points
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Range Formula
• Good for when yf = yi
x = vi,xt
y = vi,yt !1
2gt2= 0
t =2vi,y
g
x =2vi,xvi,y
g=2vi
2cos" sin"
g
x =vi2
gsin2"
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Range Formula
• Maximum for θ=45°R =vi2
gsin2!
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Example 3.5a
100 m
A softball leaves a bat with aninitial velocity of 31.33 m/s. Whatis the maximum distance one couldexpect the ball to travel?
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Example 3.6
68 m
A cannon hurls a projectile which hits a target located on acliff D=500 m away in the horizontal direction. The cannonis pointed 50 degrees above the horizontal and the muzzlevelocity is 75 m/s. Find the height h of the cliff?
h
Dθ
v0
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Relative velocity
• Velocity always defined relative to reference frame.
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Example 3.7
a) 5 mphb) 1 mph
An airport walkway moves at 3 mph. A man walks ata leisurely pace of 2 mph.
a) If he walks on the walkway in the same directionas the walkway, what is his speed as seen from theground?
b) If he walks on the walkway in the oppositedirection as the walkway, what is his speed as seenfrom the ground?
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Relative velocity in 2-d
• Sum velocities as vectors• velocity relative to
ground= velocity relative to medium + velocity of medium.
vbe = vbr + vre
boat wrtriver
river wrtearth
Boat wrtearth
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2 Cases
pointed perpendicularto stream
travels perpendicularto stream
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Example 3.8
An airplane is capable of moving 200 mph instill air. The plane points directly east, but a50 mph wind from the north distorts hiscourse.
What is the resulting ground speed?What direction does the plane fly relative tothe ground?
206.2 mph14.0 deg. south of east
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Example 3.9
An airplane is capable of moving 200 mph in stillair. A wind blows directly from the North at 50mph. The airplane accounts for the wind (bypointing the plane somewhat into the wind) andflies directly east relative to the ground.
What is the plane’s resulting ground speed?In what direction is the nose of the planepointed?
193.6 mph14.5 deg. north of east
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Chapter 4
Forces and Motion
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What is a force?
• Usually a push or pull• A Vector
Fundamental Forces:
1. Strong Nuclear2. Electromagnetic3. Weak Nuclear4. Gravity
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Newton’s First Law
• If the net force exerted on an object is zero,its velocity remains constant (both magnitudeand direction).
• Objects at rest feel no net force• Objects moving with constant velocity feel no
net force
• No net force means Σ F = 0
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Mass
• A measure of the resistance of an object tochanges in its motion due to a force
• Scalar• SI units are kg
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Newton’s Second Law
• Acceleration is proportional to net force andinversely proportional to mass.
rF! = m
ra
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Units of Force
• SI unit is Newton (N)
• US Customary unit is pound (lb)• 1 N = 0.225 lb
F = ma
1 N = 1kg !m
s2
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Gravitational Force
Weight = magnitude of Gravitational Force on anobject near the surface of the Earth
Galileo:
weight
mass
!
r a = g
" w = mg
• Weight is different on surface of other planets/moons.
• Mass is same everywhere.
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Newton’s Third Law
• Single isolated force cannot exist• For every action there is an equal and opposite
reaction• Action and Reaction Forces act on different
objects
Force on “1” due to “2”
rF12= !
rF21
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Action-Reaction Pairs
rn = ! "
rn
Fg= !F
g
'
!
r n =r " n =
Fg =
" F g =
Normal force of table holding up TV
Normal force of TV pushing down on table
Weight of TV (attracted to Earth)
Gravitational force on Earth (attracted to TV)
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Defining the Object:Free-body Diagram
• Newton’s Law uses theforces acting ON object
• n and Fg act on object• n’ and Fg’ act on other
objects
Ignore rotational motion for now.Treat object as a particle.
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Definition of Equilibrium
rF! = 0
Object is at rest or movingwith constant velocity
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Example 4.1a
A Ford Pinto is parked in a parking lot
There is no net force on the Pinto
A) TrueB) False
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Example 4.1b
A Ford Pinto is parked in a parking lot
The contact (normal) force acting on the Pinto fromthe parking lot surface ______________ .
A) Points upwardsB) Is zeroC) Points downward
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Example 4.1c
A Ford Pinto drives down a highway on the moonat constant velocity (where there is no airresistance)
The Pinto’s acceleration is __________
A) Less than zeroB) Equal to zeroC) Greater than zero
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Example 4.1d
A Ford Pinto drives down a highway on the moonat constant velocity (where there is no airresistance)
The force acting on the Pinto from the contact withthe highway is vertical.
A) TrueB) False
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Mechanical Forces
• Gravity• Normal forces• Strings, ropes and Pulleys• Friction• Springs (later)
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Rules for Ropes and Pulleys
• Force from rope points AWAY fromobject• (Rope can only pull)
• Magnitude of the force is Tension• Tension is same everywhere in the rope
• Tension does not change when goingover pulley
Approximations: Neglect mass of rope and pulley, neglect friction in pulley
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Example 4.2
I pull a 5 kg mass up with a rope, so that itaccelerates 2 m/s2. What is the tension in the rope?
T = 59 N
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Example 4.3
a) Find accelerationb) Find T, the tension abovethe bowling ballc) Find T3, the tension in therope between the pailsd) Find force ceiling must exerton pulley
a) a = g/6 = 1.635 m/s2
b) T = 57.2 Nc) T3=24.5 Nd) Fpulley=2T = 114.5 N
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Example 4.4a
2) Which statements are correct?Assume the objects are static.
T1 is _____ T2
cos(10o)=0.985 sin(10o)=0.173
A) Less thanB) Equal toC) Greater than
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Example 4.4b
2) Which statements are correct?Assume the objects are static.
T2 is ______ T3
cos(10o)=0.985 sin(10o)=0.173
A) Less thanB) Equal toC) Greater than
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Example 4.4c
2) Which statements are correct?Assume the objects are static.
cos(10o)=0.985 sin(10o)=0.173
A) Less thanB) Equal toC) Greater than
T1 is _____ Mg
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Example 4.4d
2) Which statements are correct?Assume the objects are static.
T1+T2 is ______ Mg
cos(10o)=0.985 sin(10o)=0.173
A) Less thanB) Equal toC) Greater than