Download Fnet = m a

Chapter 2
Dynamics
Kinematics, the study of how things move, can only take a
person so far. It was up to Isaac Newton, who was born the year
Galileo died, to develop what has come to be known as “classical
dynamics” which is the study of why things move as they do.
He collected three basic laws of dynamics.
1. Objects in uniform motion will stay in uniform motion
forever, unless acted upon by a net outside force.
Uniform motion means ___________________ with a
____________________
Net force means _________________________
Outside forces are the only ones that wan cause an object to change its motion.
2. Force is the cause of all change in motion
Fnet = m a
F is force measured in ________________ (or ___________ in English units)
It is the amount of _______ or _________ being exerted on an object.
m is mass measured in ________________ (or ___________ in English units)
It is the measure of the amount of ______________________.
3. For every action there is an equal but opposite reaction
This is the most difficult of the three laws to really understand so
We will put off studying this one in detail until we look at momentum
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Review of projectile motion
Let’s look at the example of a cannonball fired at certain angle above the horizontal.
According to Newton’s first law of motion and in the absence of gravity, the cannonball
should travel in a straight line forever. However, we live on Earth and there is gravity. It
acts straight downwards. Each second the cannonball remains in the air it is pulled further
and further away from it’s original path. This distance can be calculated by using “Dickie”
d = ______ + ______ + ___
where “a” is _____ m/s2.
If we neglect wind resistance the shape of the cannonball’s path is predictable. It follows
along a parabolic path.
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Mechanical Forces
Notice:
Force is a vector. It has both magnitude and direction
Force (Newtons) has sub-units (kilograms x meters / second2)
Gravity
Fg The force of gravity acting on an object. (your weight)
Fg
Fslope
=mg
The portion of the force of gravity that is directed along a slope.
(sliding force)
Fslope = m g sin
Fnormal
Fg directed perpendicular to the surface. (Your apparent weight on the slope)
Fnormal = m g cos
Friction

Ff = Fnormal 
Fno
r
q
F
g
Fslope
q
 is the coefficient of friction a measure of the stickiness between two surfaces.
It ranges between zero (no friction) and 1.0 for most objects but may be higher.
It is around 30,000 where two metal blocks have been welded together. Most 
‘s are less than 1
1. Friction always works in opposition to the net outside force.
2. There are two types of friction Static and Kinetic. 
where the materials are sliding over each other and 
there is no sliding.
3. Important: s > ke
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kinetic applies
static where
Lets Look at a Simple Elevator Problem
The elevator is resting at the bottom of a building (A). The motor turns on to give it
increasing upward speed (B). The elevator moves upward with constant speed (C). The motor
works in reverse to slow the elevator down (D) The stopped elevator rests to let the
passengers out (E). Lets fill in the graphs below.
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A) A 100 kg block is being held up by a rope.
D) A 100 kg elevator is moving upward
with a speed of 5 m/sec
B) A 100 kg elevator is accelerating
downward at 5 m/sec2 (0.5 g’s)
E) Wheezo the 100 kg cat is sliding down a
rope which will brake if he applies more
than 445 N force (about 100 lb) tension
to it. What is the minimum acceleration
he can have?
C) A 100 kg elevator is accelerating
upward at 5 m/sec2 (0.5 g’s)
F) In the problem below a small 1000 kg
car has brakes
which can supply a
braking force of about 445 N (about 100
lb). If the car is initially going 30
m/sec. What is its acc and how far will
it slide before coming to a rest?
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Pulley Problems
Pulley problems are just like the train problems only now each object applies a force to the
system because of gravity. Sometimes it moves the system forward (pos. force) and sometimes
it works against the system (neg. force). In the following problems assume that pos. is in the
direction the bigger object wants to fall.
1. Assume you have the two objects below
find the acceleration of the system.
3. Now that you have found the
acceleration of the system find the
tension in the rope.
pulley
20 kg
20 kg
30 kg
30 kg
pulley
a = _________ m/s2
2.
. Frope = ___________ N
Assume you have the two objects below
find the acceleration of the system.
4. Now that you have found the
acceleration of the system find the
tension in the rope.
a = _________ m/s2
Frope = ___________ N
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Pulley Lab
Comparing theory to reality
Set up a pulley to your data acquisition equipment. You will be measuring the acceleration of
systems when the masses on each side are not equal.
Mass A
Mass B
Average Acceleration
( + or - )
250 gr
200 gr
_____________
____________
450 gr
400 gr
_____________
____________
______gr
_______gr
_____________
____________
______gr
_______ gr
_____________
____________
Now lets compute the theoretical acceleration that we would have predicted
Mass A
Mass B
Theoretical Acceleration
250 gr
200 gr
____________
450 gr
400 gr
____________
gr
gr
____________
gr
gr
_____________
Explain your differences.
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Multiple bodies
In the pictures below complete the force diagram and calculate the force. (Assume plus is
forward)
1. A model railroad train accelerates at a
rate of 2.0 m/sec2 What net force must
the engine exert?
60 kg
2. Find the tension in the first coupler if
the train is accelerating at 2.0 m/sec2?
3. Find the tension in the second coupler if
the train is accelerating at 2.0 m/sec2?
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60 kg
100 kg
60 kg
60 kg
100 kg
60 kg
60 kg
100 kg
You are pushing the blocks below over a
frictionless surface. What force must
you exert to maintain an acc. of 2.0
m/sec2?
Fapp
100 kg
5. What is the force the 100 kg block
exerts on the 20 kg block?
20 kg
60 kg
6. What is the force the 20 kg block
exerts on the 60 kg block?
100 kg
20 kg
Fapp
60 kg
100 kg
20 kg
4.
60 kg
Fapp
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Slope Problems
In the pictures below complete the force diagram for a 100 kg block. and calculate the force. (Assume plus is
forward down the hill)
1. Find the acceleration of the block and its
Fapp = m g cos 
Fnet = m a
Fslope = m a
Fapp
 =ma

a = ________
Fapp = ___________
Fno
r
q
F
g
q
Fslope
As the slope becomes steeper the acceleration becomes greater and the object’s apparent weight becomes less.
Calculate each of these for the following angles.
Angle of slope
=0
Acceleration
Apparent weight
 = 10
 = 20
 = 30
 = 40
 = 50
 = 60
 = 70
 = 80
 = 90
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From the information derived on the previous page, let us graph the relative accelerations and apparent
weights as the steepness of the slope is changed.
Calculating Friction
To the right are three identical triangularly shaped
blocks. Each is being pulled along the ground.
Which do you think is hardest to drag?
___________________________
_____________________________________________________________________________
__
Scientists did a lot of studying of friction
and determined that friction is directly
related to two things. The force between
objects (related to the object’s apparent
weight) and the stickiness of the materials
involved
Ffriction = ( Fnormal )  general equation
Ffriction = ( m g ) 
on flat ground
Ffriction = ( m g cos ) 
on slopes
_______________________________________________________________________________
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
Problems with Friction
Rubber tires on asphalt have a coefficient friction of 0.65 If you and your car (1200 kg) are
moving along the road at 30 m/sec and you lock the brakes, how far will you slide?
Picture below
First find the acceleration of the car.
Fnet = m a
- FFrict = m a
- m g  = m a
a = g  = _______________________
Now knowing the acceleration we find the distance the car travels
vf2 = vo2 + 2 a ( d )
0 = 302 + 2 ______ ( d )
d
=
_______________________
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It turns out that there are two types of friction Kinetic (sliding) and Static (not sliding) Static
friction is always greater than kinetic friction. This is why we have anti-lock braking
systems, to prevent tires from sliding. In the case of the car above the static friction between
tires and road is 0.99.
Old fashioned locking brakes
Initial
velocity
10 m/sec
20 m/sec
30 m/sec
40 m/sec
50 m/sec
Acceleration
when sliding
New antilock brakes
Distance to
stop
Acceleration
with antilock
Distance to
stop
Graph both stopping curves. Do anti-lock brakes make a substantial difference?
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Lets see what happens when you try to slow down a car while going down a 30 degree slope
while having an initial speed of 30 m/sec. Assume that the coefficient of friction is 0.99 with
our anti-lock brakes.
First we must complete the vector diagram for the picture below.
Givens:
Fno
r
Fslope
q
F
g
q
A. Find the acceleration of the car.
B. Find the distance the car travels as it comes to a stop.
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Putting on the brakes while going up a hill is a different story. Assume we are traveling up a
30 degree hill with an initial speed of 30 m/sec. Given a coefficient of friction of 0.99 we want
to find the distance it takes to come to a stop.
Givens:
Fno
r
Fslope
q
F
g
q
C. Find the acceleration of the car.
D. Find the distance the car travels as it comes to a stop.
Review: Stopping distance for a car with initial velocity of 30 m/sec
Up hill
__________________
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On the flat __________________
Down hill
__________________
1. An object is stationary on a horiz. surface (  = 0.9) A force is applied as shown.
Complete the freebody diagram then calculate the force of friction.
2. An object is sliding on a horiz. surface with a speed of 5 m/sec when you first see it.
There is friction ( = 0.9) Complete the freebody diagram then calculate the
acceleration of the block.
3. An object is sliding up an inclined surface with a speed of 5 m/sec when you first see it.
Complete the freebody diagram then calculate the acceleration of the block. There is no
friction. The angle is 30 degrees. Find acceleration of the block.
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4. An object is sliding up an inclined surface with a speed of 5 m/sec when you first see it.
There is friction ( = 0.9) Complete the freebody diagram then calculate the
acceleration of the block. The angle is 30 degrees.
5. An object is sliding down an inclined surface with a speed of 5 m/sec when you first see
it. There is friction ( = 0.9) Complete the freebody diagram then calculate the
acceleration of the block. The angle is 30 degrees.
6. An object is stationary on an inclined surface. There is friction ( = 0.9) Complete the
freebody diagram then calculate the force of friction on the block. The angle is 30
degrees.
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7. In the pulley system below find the acceleration of the blocks. Show a complete free
body diagram for doing so)
8. In the pulley system below find the acceleration of the blocks. Show a complete free
body diagram for doing so)
9. In the pulley system below find the acceleration of the blocks. The smaller block
experiences friction ( = 0.9) Show a complete free body diagram for doing so)
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How the Coefficient of Friction
Is Determinied
To determine the coefficient of friction, we drag
known weight (mg) along the floor at a constant
measure the force of pull. It will be equal in
to the force of friction. We then substitute into
g ) 

a mass of
speed and
magnitude
Ffrict = ( m
linoleum floor. If you need to pull with a force of 44.5 Newtons (10 lbs) what is the coefficient
of friction?
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Putting it together
1. Our friend Fred the cat is car moving at 30 m/s. He sees a beautiful feline 40 meters (about
120 feet) away who is hitching a ride. He slams on the brakes ( = 0.6) Will he stop before he
reaches her or will he run her over?
____________________________
2. In the problem above. How many g’s is Fred pulling as he comes to a stop?
____________________________
3. Two things limit a car’s movement up a hill. One is the power of the engine and the other is
the friction between the tires and the road. Let us assume that our car has a very powerful
motor. So the limiting factor will be friction with the ground. What is the steepest hill
(degrees) that a car can drive up at a constant speed of 30 m/sec. if  = 0.9
____________________________
4. A 10 kg cat is suspended from the ceiling by a thin rope. If the cat is stationary, what is the
tension in the rope?
____________________________
5. Does he have acceleration?
____________________________
6. Is the acc 0, positive, or negative? ____________________________
7. Is gravity acting on him?
____________________________
8. The moment the rope breaks (bummer), does he have an initial velocity?
____________________________
9. Does he have an initial acceleration? ___________________________
10. How much time elapses until he hits the tank of piranha fish which is 3 meters below?
____________________________
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Pulleys with masses on slopes and with friction
Given an 80 kg block on a 40 degree slope (with respect to the horizontal) which has a
coefficient of 0.4. the block is attached to a pulley at the top of the slope. On the other side
of the pulley hangs vertically a 200 kg block .
A. Draw a complete picture of the situation.
B. Find the direction of the net force on the 80 kg block if friction is not included.
C. Which way will the force of friction be acting? ___________________
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D. Include friction and calculate the acceleration of the system.
E. Find the tension in the rope. Include a new picture.
Quick review questions
1. A zoo keeper devises a rubber-band gun to shoot food to a monkey who is too shy to come
down from the trees. If the monkey does not move, should the keeper aim above, at, or below
the monkey?
A. ABOVE THE MONKEY
B. AT THE MONKEY
C. BELOW THE MONKEY
2. If in the problem above the monkey lets go the instant
the banana is fired, if he wants to hit the monkey
he should aim
A. ABOVE THE MONKEY
B. AT THE MONKEY
C. BELOW THE MONKEY
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3. You have two blocks each with a weight of 100 Newtons. (hint: not mass but
weight) In the first situation you hang a 100 N block over the pulley, in the
second situation you just pull with a force of 100 Newtons. (neglect frict.)
A. The acceleration in the second case is greater
B. The acceleration in the first case is greater
C. The acceleration is the same in both cases
4. In deep space (no gravity) a pellet is fired into a spiral tube. When it
emerges, what direction will it travel?
5. Two smooth balls of exactly the same size, one made of wood and the other of iron, are
dropped from a high building to the ground below. Because the iron ball has a greater inertia its
acceleration will be (neglect wind resistance)
A.
B.
C.
D.
greater than that of the wooden ball
less than that of the wooden ball
the same as that of the wooden ball
there is not enough information to determine this
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6. If blocks A and B each have a weight
100 Newtons and they are initially
at rest, the tension in the rope is
A.
B.
C.
D.
E.
Zero Newtons
0 N. < tension < 100 N.
100 N.
100 N. < tension < 200 N.
200 N.
7. In the problem above, everything is the same but the blocks are initially moving
with a speed of 2 m/sec. Now the tension in the rope is
A.
B.
C.
D.
E.
Zero Newtons
0 N. < tension < 100 N.
100 N.
100 N. < tension < 200 N.
200 N
8. What if the blocks above are not of equal weight but block A is 50 Newtons and block
B is 100 Newtons. Assuming they start from rest and there is no friction. What
is the tension in the rope?
A.
B.
C.
D.
E.
Zero Newtons
0 N. < tension < 100 N.
100 N.
100 N. < tension < 200 N.
200 N
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