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First Exam Coming Up!
• Sunday, 12 October 6:10 – 7:30 PM.
• Locations to be posted online.
• Yes this is a Sunday!
• There will be 17 questions on exam.
• If you have a legitimate conflict, you must ask Prof.
Shapiro by Oct. 8 for permission to take the make-up
exam. Email him the reason for missing the exam,
and your schedule for Oct. 15 – 18.
Newton’s Laws of Motion
• Components of Equation of Motion
vx = v0x + ax t vy = v0y + ay t
x = x0 + v0xt + ½ ax t2 y = y0 + v0yt + ½ ay t
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• Projectile Motion Superposition of two motions:
Constant vx and free-fall vy
Then solve for : y = f(x) or trajectory:
• Relative Velocity: (e.g., cars on highway, airplane in wind, boat on river, …)
Fixed Frame: A Moving Frame: B Moving Object: P
g
vh oo
2
sin22
g
vR oo 2sin2
h
x
y
R
ov
oyv
oxv
ABBPAP vvv ///
TOPICS FROM PREVIOUS LECTURE (motion in 2-D)
The change in
velocity for the ball
during the bounce is
Before After
3 m/s
2 m/s
a) 1 m/s downward
b) 2 m/s upward
c) 1 m/s upward
d) 5 m/s downward
e) None of the
above
i-Clicker
Between point A and B, the
direction of the average
acceleration for this
motorcyclist is
a) b) c) d) e)
i-Clicker
i-Clicker
h
x
y
R
ov
oyv
oxv
ax =0 ; ay = -g
NEWTON’S LAWS OF MOTION So Far: DESCRIPTION of motion in terms of position vector, velocity, acceleration. (KINEMATICS) NOW: What is the CAUSE of motion? (DYNAMICS) “Laws of Motion” formulated by Newton over 300 years ago
Forces and interactions
Common experience: Force is a “push” or a “pull” Force is a vector Different types of forces:
• Contact Force (kick soccer ball, push book, tow car, … )
• Long range forces, act through “empty” space (electric force, magnetic force, gravitational force, … )
Unit of force in SI units: Newton [N] We will focus on contact forces and gravity.
pull push
Superposition of forces
Forces are vectors add to give resultant. NET FORCE: EQUILIBRIUM: Net force = 0
2F
1F
R
xR
yR
R
...321 FFFFR
...321 xxxxx FFFFR ...321 yyyyy FFFFR
222
zyx RRRR
?0 F
0 ;0 ;0 zyx FFF
3F
2F
1F ! 0 F
xF3
yF3
Newton’s first law Ancient Idea: The natural state of an object is at rest. If an object moves, it’s because a force acts on it.
NEWTON: The nature of a body is to resist change in motion.
Newton’s First Law When no net force acts on an object, its acceleration is zero An isolated object with no net force acting on it is either at
rest, or is moving with a constant velocity.
INERTIA: The tendency of a body to resist change in .
INERTIAL FRAMES OF REFERENCE An inertial frame of reference is one in which Newton’s 1st Law is valid (The frame is not accelerating!!) An elevator moving at constant velocity is an inertial frame. An accelerating elevator is NOT an inertial frame.
Ax
0 then 0 If aF
v
Az
Ay
Bx
Bz
By
ABv /
Inertial frame
Also inertial frame if constant. / ABv
Copyright © 2012 Pearson Education Inc.
Newton’s Second Law
• If the net force on an object is not zero, it causes the
object to accelerate.
Copyright © 2012 Pearson Education Inc.
Force and acceleration
• The acceleration of an object
is directly proportional to the
net force (sum of all the
forces) on the object.
• Conversely, the net force is
proportional to the
acceleration.
• The proportionality constant
is called the mass of the
object.
Newton’s Second Law When the net force causes the object to accelerate in the same direction as the net force. The net force is directly proportional to . The proportionality constant is the mass of the object.
Mass and Force:
• Force changes motion of an object • Mass (inertia) resists change in motion
Mass ≠ weight
• Weight is the force on a body due to gravity. • Weight is proportional to mass. • The mass of a body is the same everywhere in the world. • The weight of a body depends on where you are (earth?
moon?)
a
F
0 F
amF
scalar vector
Statement Of Newton’s Second Law The acceleration of an object is proportional to the net force acting on that object and inversely proportional to its mass.
or
Unit of force: Newton [N]
1 N = (1 kg) X (1 m/s2) = 1kg m/s2 EXAMPLE: A block (m = 5.0 kg) is at rest at t = 0. Then, a constant force Fx acts on it. At t = 3.0 s, the block has moved 2.3 m. Find Fx.
Have: m, need ax. But, also have t and (x - xo)
xx ma F
m
Fa
maF
zzyyxx maFmaFmaFamF ; ;
Fx
ax
m
2
21)( tatvxx xoo
2
22m/s 51.0
)s 0.3(
)m 3.2(2)(2
t
xxa o
x
N 6.2m/skg 6.2)m/s 51.0)(kg 0.5( 22 x F
Mass And Weight
Weight is the attractive force, , exerted by the earth’s gravity on an object. = weight of object = Because and then in free fall (i.e., no force other than gravity acting on object) we have So we can compare masses by measuring weight at earth’s surface. Note:
• Weight is a force that acts on a body at all times • varies with location, we will use g = 9.8 m/s2. • Avoid incorrect usage (i.e. an object’s weight is 2 kg) • English unit for weight: 1 Pound = 4.44822 Newtons • English unit for mass: 1 Pound = 0.45359 kg
Measuring forces: Convenient to use elongation of a spring. (Elongation of a spring is proportional to the force exerted on it.)
gF
amF
gF
w
gmw
ga
g
Six identical curling stones are pushed horizontally along the ice. For each stone, the instantaneous velocity and acceleration are given. Assume the ice is frictionless. Rank the stones based on the magnitude of the force exerted on them.
a) C = F > A = D > B = E
b) B = D = F > A = C = E
c) F > C > D > A > B > E
d) The magnitude of the force is
equal and non zero for all.
e) The magnitude of the force is
zero for all.
i-Clicker
i-Clicker
mg
T
0 F
Elevator moves, but at constant velocity a = 0
F = ma, therefore in particular Therefore, T – mg = 0 ; T = mg.
0 yF
An object moves along a straight path from its starting point to a second point, and then returns to its starting point. The graph of the speed of the object as a function of time is shown to the right.
What would a plot of the force versus time
graph look like?
a) It will be constant, because the slope of the
velocity never changes.
b) It will be “kick” in the positive direction at
t=2 and be zero everywhere else.
c) It will be a “kick” in the negative direction at
t=2 and be zero everywhere else.
d) It will be positive from t=0 to t=2 and
negative from t=2 to t=4.
e) It will be negative from t=0 to t=2 and
positive from t=2 to t=4.
i-Clicker
Newton’s Third Law Whenever two objects interact, the forces that the objects exert on each other are always equal in magnitude and opposite in direction. Restated: If object A exerts a force on B (“action”) then B exerts an equal and opposite force on A (“reaction”). These two forces:
• Have the same magnitude • Are opposite in direction • Act on different bodies!!!!!
Suppose two balls collide “action-reaction pair” Forces exist in pairs, acting on different objects (e.g., bat hits ball; hammer hits nail; foot kicks butt … )
A B
Bon A F
Aon BF
Bon A Aon B FF
Consider a box on a table on the earth Simpler to understand if consider forces on box alone: Free Body Diagrams: an aid to solving problems
• applied to one object with mass m (or a group moving as one). • Use only forces applied to this body • Draw diagram for each body separately
wFg
'n
n
(normal force: force of table
on box)
Reaction to (force of box
on table)
n
n
w
Box is at rest, so:
0 F
0 xF
0 yF
x: no forces
y: n – mg = 0 n = mg
and are
NOT action/reaction pair.
They act on same object.
and
ARE action/reaction pair
n
w
n
'n
amF
gF
Reaction to weight of box
Copyright © 2012 Pearson Education Inc.
Applying Newton’s Third Law
Newton’s apple rests on a table. Identify
the forces that act on it and the action-
reaction pairs.
20
A baseball is thrown from right field to home plate (HP),
traveling from right to left in the diagram.
If we ignore air friction, which is the correct free-body
diagram for the baseball at point T?
i-Clicker
EXAMPLE: Apparent contradiction – a horse pulling a cart. If the force of horse on cart equals the force of cart on horse how does the cart move???? Isolate cart and draw free body diagram At the instant the cart moves, it accelerates: 1st law: so is NOT constant (acceleration) 2nd law: 3rd law: Note that , are NOT reaction pair!!!
n
w
amF
m
FaFmaF HC
xHCxx
0 wnFy
x
n
w
0 F
Con HF
Hon CF
Con HF
n
w Con HF
y
v
EXAMPLE: A skier of mass 65.0 kg is pulled by a tow rope up a snow-covered ski slope at a constant speed. The rope and the slope are inclined by 26o. Draw a free body diagram and calculate the tension in the tow rope. moving, but no acceleration consider components in tilted coordinate system:
0 F
)26cos(0 wnFy
T
n
w
26o
T
x y
26o
wsin(26o)
)26sin(0 wTFx
)26sin()26sin( mgwT
T = (65.0 kg)*(9.80 m/s2)*sin(26o)
T = 279 N
i-Clicker
MILK
mCg nC
FT on C
nT
mTg
FC on T FH on T
action/reaction pair
Ton CCon T FF
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First Exam Coming Up!
• Sunday, 12 October 6:10 – 7:30 PM.
• Locations to be posted online.
• Yes this is a Sunday!
• There will be 17 questions on exam.
• If you have a legitimate conflict, you must ask Prof.
Shapiro by Oct. 8 for permission to take the make-up
exam. Email him the reason for missing the exam,
and your schedule for Oct. 15 – 18.
