Transcript Lecture05-09

Chapter 4 – Kinematics (cont.)
Recap:
Dropping a Package
You drop a package
from a plane flying at
a) quickly lag behind the plane
while falling
constant speed in a
b) remain vertically under the
plane while falling
straight line. Without
air resistance, the
package will:
c) move ahead of the plane while
falling
d) not fall at all
Dropping a Package
You drop a package
from a plane flying at
constant speed in a
straight line. Without
air resistance, the
package will:
a) quickly lag behind the plane
while falling
b) remain vertically under the
plane while falling
c) move ahead of the plane while
falling
d) not fall at all
Both the plane and the package have
the same horizontal velocity at the
moment of release. They will maintain
this velocity in the x-direction, so they
stay aligned.
Follow-up: what would happen if air resistance is present?
A battleship simultaneously fires two shells at two
enemy submarines. The shells are launched with the
same magnitude of initial velocity. If the shells follow
the trajectories shown, which submarine gets hit first ?
a
b
c) both at the same time
A battleship simultaneously fires two shells at two
enemy submarines. The shells are launched with the
same magnitude of initial velocity (i.e., speed). If the
shells follow the trajectories shown, which submarine
gets hit first ?
The flight time is fixed by the
motion in the y-direction. The
higher an object goes, the longer it
stays in flight. The shell hitting
submarine #2 goes less high,
therefore it stays in flight for less
time than the other shell. Thus,
submarine #2 is hit first.
a
b
c) both at the same time
Follow-up: Did you need to know that they had the same initial speed?
Range: the horizontal distance a projectile travels
As before, use
(y = 0 at landing)
and
Eliminate t and solve
for x when y=0
Sin 

Range is maximum at 45o
sin(2 x 45o) = sin(90o) = 1
Range Gun
If the range gun launches
a ball 6 meters with a
launch angle of 45
degrees, at which of
these angles should a
ball be launched to
land in a bucket at 3
meters?
a) 10 degrees
b) 22.5 degrees
c) 60 degrees
d) 75 degrees
e) one would also need the
launch velocity of the range
gun to know
Range Gun
If the range gun launches a ball 6
meters with a launch angle of
45 degrees, at which of these
angles should a ball be
launched to land in a bucket
at 3 meters?
a) 10 degrees
b) 22.5 degrees
c) 60 degrees
d) 75 degrees
e) one would also need the launch
velocity of the range gun to know
The range is proportional to sin2θ, so to travel half the distance,
the ball would need to be launched with sin2θ = 0.5.
sin(75o) = 0.5, so θ=75o
Note: there are two angles that would work [sin(30o) = 0.5 also].
How are the two solutions different?
Symmetry in projectile motion
Dropping the Ball III
A projectile is
launched from the
ground at an angle of
30°. At what point in its
trajectory does this
projectile have the
least speed?
a) just after it is launched
b) at the highest point in its flight
c) just before it hits the ground
d) halfway between the ground and the
highest point
e) speed is always constant
Dropping the Ball III
A projectile is
launched from the
ground at an angle of
30º. At what point in its
trajectory does this
projectile have the
least speed?
The speed is smallest at the
highest point of its flight
path because the
y-component of the velocity
is zero.
a) just after it is launched
b) at the highest point in its flight
c) just before it hits the ground
d) halfway between the ground and the
highest point
e) speed is always constant
Punts I
Which of the
three punts
h
has the
longest hang
time?
a
b
d) all have the same hang time
c
Punts I
Which of the
three punts
h
has the
longest hang
time?
a
b
d) all have the same hang time
The time in the air is determined by the vertical motion!
Because all of the punts reach the same height, they all
stay in the air for the same time.
Follow-up: Which one had the greater initial speed?
c
•• On a hot summer day, a young girl swings on a rope above
the local swimming hole. When she lets go of the rope her initial
velocity is 2.25 m/s at an angle of 35.0° above the horizontal. If
she is in flight for 0.616 s, how high above the water was she when
she let go of the rope?
Time to hit the water: t=0.616
Initial velocity: 2.25 m/s at 35o above the horizontal
y = (v0 sinθ) t - 1/2 g t2
y = 2.25 m/s * sin(35o) * (0.616 s) - 1/2 (9.8 m/s2) (0.616 s)2
= - 1.07 m
At time t= 0.616 s, the girl is 1.07 m below her starting position,
so her initial position was 1.07m above the water.
In a friendly game of handball, you hit the ball essentially at ground level and send it
toward the wall with a speed of 14 m/s at an angle of 34° above the horizontal.
(a) How long does it take for the ball to reach the wall if it is 3.8 m away?
(b) How high is the ball when it hits the wall?
In a friendly game of handball, you hit the ball essentially at ground level and send it
toward the wall with a speed of 14 m/s at an angle of 34° above the horizontal.
(a) How long does it take for the ball to reach the wall if it is 3.8 m away?
(b) How high is the ball when it hits the wall?
d = 3.8 m
v0 = 14 m/s
θ = 34o
v0x = (14 m/s) cos(34o) = 11.6 m/s
v0y = (14 m/s) sin(34o) = 7.83 m/s
(a) d = v0x t
t = d/v0x = (3.8m) / (11.6 m/s) = 0.33 s
(b) h = v0y t - 1/2 g t2
= (7.83 m/s) (0.33s) -1/2 (9.8 m/s2) (0.33 s)2 = 2.0 m
In a friendly game of handball, you hit the ball essentially at ground level and send it
toward the wall (3.8 m away) with a speed of 14 m/s at an angle of 34° above the
horizontal.
(c) what are the magnitude and direction of the ball's velocity when it strikes the wall?
(d) Has the ball reached the highest point of its trajectory at this time? Explain.
d = 3.8 m v0 = 14 m/s θ = 34o
v0x = (14 m/s) cos(34o) = 11.6 m/s
v0y = (14 m/s) sin(34o) = 7.83 m/s
t = d/v0x = (3.8m) / (11.6 m/s) = 0.33 s
vy = v0y - g t = 7.83 m/s - (9.8m/s2) (0.33s) = 4.6 m/s
vx = v0x
The vertical component of
velocity is still positive, that
is, the ball is still going up.
So the ball has not yet
reached its highest point.
Kinematics: Assumptions,
Definitions and Logical Conclusions
•
•
•
•
What have we done so far?
Defined displacement, velocity, acceleration (also
position, distance, speed)...
Defined scalers (like speed) and vectors (like velocity)
Laid out assumptions about free-fall
noticed that 2-dimensional motion is really just two,
simultaneous, 1-dimensional motions.
Used this to shoot a monkey, range
out a small cannon, etc.
This wasn’t physics. This was preparing the
language needed to talk about physics.
Chapter 5
Newton’s Laws of Motion
(sections 5.1-5.4)
Newton’s Laws
How can we consistently and generally
describe the way objects move and interact?
Isaac Newton
1643-1727
Newton in a 1702
1689 portrait by
Godfrey Kneller
Nature and nature's laws lay hid in night;
God said "Let Newton be" and all was light.
Newton’s epitaph
Alexander Pope (English poet, 1688-1744)
I do not know what I may appear to the world, but to myself I
seem to have been only like a boy playing on the sea-shore,
and diverting myself in now and then finding a smoother
pebble or a prettier shell than ordinary, whilst the great ocean
of truth lay all undiscovered before me.
from a memoir by Newton
Force
Force: push or pull
Force is a vector – it has magnitude and
direction
Mass
Mass is the measure of
how hard it is to change
an object’s velocity.
Mass can also be thought
of as a measure of the
quantity of matter in an
object.
Newton’s First Law of Motion
If you stop pushing an object, does it stop
moving?
Only if there is friction!
In the absence of any net external force, an object at rest
will remain at rest.
In the absence of any net external force a moving object
will keep moving at a constant speed in a straight line.
This is also known as the Law of Inertia.
Inertia
Newton’s First Law
A hockey puck
slides on ice at
constant velocity.
What is the net
force acting on
the puck?
a) more than its weight
b) equal to its weight
c) less than its weight but more than zero
d) depends on the speed of the puck
e) zero
Newton’s First Law
A hockey puck
slides on ice at
constant velocity.
What is the net
force acting on
the puck?
a) more than its weight
b) equal to its weight
c) less than its weight but more than zero
d) depends on the speed of the puck
e) zero
The puck is moving at a constant velocity, and
therefore it is not accelerating. Thus, there must be
no net force acting on the puck.
Follow-up: Are there any forces acting on the puck? What are they?
Newton’s First Law
You put your book on
a) a net force acted on it
the bus seat next to
b) no, or insufficient, net force acted on it
you. When the bus
c) it remained at rest
stops suddenly, the
book slides forward off
the seat. Why?
d) it did not move, but only seemed to
e) gravity briefly stopped acting on it
Newton’s First Law
You put your book on
a) a net force acted on it
the bus seat next to
b) no, or insufficient, net force acted on it
you. When the bus
c) it remained at rest
stops suddenly, the
book slides forward off
the seat. Why?
d) it did not move, but only seemed to
e) gravity briefly stopped acting on it
The book was initially moving forward (because it was on
a moving bus). When the bus stopped, the book
continued moving forward, which was its initial state of
motion, and therefore it slid forward off the seat.
Calibrating force
Two equal weights exert twice the force of one; this
can be used for calibration of a spring:
Experiment: Acceleration vs Force
Now that we have a calibrated spring, we can do
more experiments.
Acceleration is proportional to force:
Experiment: Acceleration vs Mass
Acceleration is inversely proportional to mass:
Newton’s Second Law of Motion
Acceleration is proportional to force:
Acceleration is inversely proportional to mass:
Combining these two observations gives
Or, more familiarly,
Newton’s Second Law of Motion
An object may have several forces acting on it; the
acceleration is due to the net force:
SI unit for force Newton is defined using this equation as:
1 N is the force required to give a mass of 1 kg an
acceleration of 1 m/s2
Units of force: Newtons
Force of Gravity
The weight of an object is the force acting on it due to gravity
Weight: W = Fg = ma = mg vertically downwards
Since
, the weight of an object in Newtons is
approximately 10 x its mass in kg
adult human
70 kg
700 N ~ 160 lbs.
There is no “conversion” from kg to pounds!
(Unless you specify what planet you are assuming)
Newton’s First and Second Laws
(I)
In order to change the velocity of an object –
magnitude or direction – a net force is required.
(II)
What about the bus... From the perspective of
someone who didn’t know they were on the bus?
Inertial Reference Frames
(I)
In order to change the velocity of an object –
magnitude or direction – a net force is required.
(II)
Newton’s First and Second Laws do not
work in an accelerating frame of reference
An inertial reference frame is one in which the
first and second laws are true. Accelerating
reference frames are not inertial.
Was the bus an inertial reference frame?
Is the earth an inertial reference frame?
No, but acceleration due to earth’s rotation around
Its axis (0.034 m/s2), and due to earth’s rotation around sun
(smaller) are negligible compared to g; so approximately yes.
Analyzing the forces in a system
Free-body diagrams:
A free-body diagram shows every force acting on
an object.
• Sketch the forces
• Isolate the object of interest
• Choose a convenient coordinate system
• Resolve the forces into components
• Apply Newton’s second law to each coordinate
direction
Free-body Diagram
Example of a free-body diagram:
Newton’s First Law
A book is lying at
rest on a table.
The book will
remain there at
rest because:
a) there is a net force but the book has too
much inertia
b) there are no forces acting on it at all
c) it does move, but too slowly to be seen
d) there is no net force on the book
e) there is a net force, but the book is too
heavy to move
Newton’s First Law
A book is lying at
rest on a table.
The book will
remain there at
rest because:
a) there is a net force but the book has too
much inertia
b) there are no forces acting on it at all
c) it does move, but too slowly to be seen
d) there is no net force on the book
e) there is a net force, but the book is too
heavy to move
There are forces acting on the book, but the only
forces acting are in the y-direction. Gravity acts
downward, but the table exerts an upward force
that is equally strong, so the two forces cancel,
leaving no net force.
Newton’s Third Law of Motion
Forces always come in pairs, acting on different
objects:
If object 1 exerts a force
object 2 exerts a force –
on object 2, then
on object 1.
These forces are called action-reaction pairs.
Some action-reaction pairs
Action-reaction pair?
a) Yes
b) No
Newton’s 3rd: F12 = - F21
action-reaction pairs are equal and
opposite, but they act on different bodies
Newton’s Third Law of Motion
Although the forces are the same, the
accelerations will not be unless the objects
have the same mass.
Q: When skydiving, do you
exert a force on the
earth? Does the earth
accelerate towards you?
Is the magnitude of the
acceleration of the earth
the same as the magnitude
of your acceleration?
Newton’s Third Law of Motion
Contact forces:
The force exerted by one
box on the other is
different depending on
which one you push.
Assume the mass of the
two objects scales with
size, and the forces
pictured are the same.
In which case is the
magnitude of the force
of box 1 on box 2
Truck on Frozen Lake
A very large truck sits on a
frozen lake. Assume there
is no friction between the
tires and the ice. A fly
suddenly smashes against
the front window. What
will happen to the truck?
a) it is too heavy, so it just sits there
b) it moves backward at constant speed
c) it accelerates backward
d) it moves forward at constant speed
e) it accelerates forward
Truck on Frozen Lake
A very large truck sits on a
frozen lake. Assume there
is no friction between the
tires and the ice. A fly
suddenly smashes against
the front window. What
will happen to the truck?
a) it is too heavy, so it just sits there
b) it moves backward at constant speed
c) it accelerates backward
d) it moves forward at constant speed
e) it accelerates forward
When the fly hit the truck, it exerted a force on the truck
(only for a fraction of a second). So, in this time period, the
truck accelerated (backward) up to some speed. After the
fly was squashed, it no longer exerted a force, and the truck
simply continued moving at constant speed.
Follow-up: What if the fly takes off, with the same speed in the
direction from whence it came?
Contact Force
Two blocks of masses 2m and m
a) 2F
are in contact on a horizontal
b) F
frictionless surface. If a force F0
c) ½F
is applied to mass 2m, what is the
d) 1/3F
force on mass m ?
e) ¼F
F
2m
m
Contact Force
Two blocks of masses 2m and m
a) 2F
are in contact on a horizontal
b) F
frictionless surface. If a force F0
c) ½ F
is applied to mass 2m, what is the
d) 1/3 F
force on mass m ?
e) ¼ F
The force F0 leads to a specific
acceleration of the entire system. In
order for mass m to accelerate at the
same rate, the force on it must be
smaller! For the two blocks together,
F0 = (3m)a. Since a is the same for both
blocks, Fm = ma
F
2m
m
A 71-kg parent and a 19-kg child meet at the center of an ice rink. They
place their hands together and push.
(a) Is the force experienced by the child more than, less than, or the
same as the force experienced by the parent?
(b) Is the acceleration of the child more than, less than, or the same as
the acceleration of the parent? Explain.
(c) If the acceleration of the child is 2.6 m/s2 in magnitude, what is the
magnitude of the parent’s acceleration?
On vacation, your 1300-kg car pulls a 540-kg trailer away from a
stoplight with an acceleration of 1.9 m/s2
(a) What is the net force exerted by the car on the trailer?
(b) What force does the trailer exert on the car?
(c) What is the net force acting on the car?
An archer shoots a 0.022-kg arrow at a target with a speed of 57 m/s.
When it hits the target, it penetrates to a depth of 0.085 m.
(a) What was the average force exerted by the target on the arrow?
(b) If the mass of the arrow is doubled, and the force exerted by the
target on the arrow remains the same, by what multiplicative factor
does the penetration depth change? Explain.