Download Chapter 4 Dynamics: Newton`s Laws of Motion

Chapter 4
Dynamics: Newton’s Laws of
Motion
• Force
• Newton’s First Law of Motion
• Mass
• Newton’s Second Law of Motion
• Newton’s Third Law of Motion
• Weight – the Force of Gravity; and the Normal Force
• Applications Involving Friction, Inclines
Recalling Last Lectures
Relative Velocity
Consider a person walking on
the boat with velocity
relative to the boat.
His velocity relative to the shore will be
given by:
using:
(3.19)
Then
Relative Velocity
Notes:
In general, if
is the velocity of A relative to B, then the velocity of B relative to A,
will be in the same line as
but in opposite direction:
(3.20)
If
is the velocity of C relative to A, then
be given by:
(3.21)
, the velocity of C relative to B, will
Relative Velocity
Problem 3.37 (textbook) : Huck Finn walks at a speed of 0.60 m/s across his raft
(that is, he walks perpendicular to the raft’s motion relative to the shore). The raft is
travelling down the Mississippi River at a speed of 1.70 m/s relative to the river bank
(Fig. 3–38). What is Huck’s velocity (speed and direction) relative to the river bank?
Solution developed on the blackboard
Relative Velocity
37. Call the direction of the flow of the river the x direction, and the direction of Huck
walking relative to the raft the y direction.
r
v H uck
rel. b ank
r
r
= v H uck + v raft rel. = ( 0, 0.60 ) m s + (1.70, 0 ) m s
rel. raft
b ank
= (1.70, 0.60 ) m s
r
v Huck
r
v Huck
rel. bank
rel. raft
Magnitude: v Huck
=
θ
r
v
ra ft
r e l. b a n k
(c u r r e n t )
1.70 2 + 0.60 2 = 1.80 m s
rel. bank
Direction: θ = tan
−1
0.60
1.70
= 19 o relative to river
Relative Velocity
Problem 5.50 (textbook) : An unmarked police car, traveling a constant 95 Km/h is
passed by a speeder traveling 145 Km/h. Precisely 1.00 s after the speeder passes,
the policeman steps on the accelerator. If the police car’s acceleration is 2.00 m/s2.
How much time elapses after the police car is passed until it overtakes the speeder
(assumed moving at constant speed)?
Solution developed on the blackboard
Relative Velocity
52. Take the origin to be the location at which the speeder passes the police car, in the
reference frame of the unaccelerated police car.
The speeder is traveling at


= 40.28 m s

 3.6 km h 
145 km h 
1m s
relative to the ground, and the policeman is traveling at
 1m s 
= 26.39 m s

 3.6 km h 
95 km h 
relative to the ground.
Relative to the unaccelerated police car, the speeder is traveling at
13.89 m s = vs
and the police car is not moving.
Relative Velocity
Do all of the calculations in the frame of reference of the unaccelerated police car.
The position of the speeder in the chosen reference frame is given by
∆xs = vs t
The position of the policeman in the chosen reference frame is given by
∆x p = a p ( t − 1) , t > 1
1
2
2
(Note that he policeman does not move relative to the reference frame until he starts
accelerating his car).
The police car overtakes the speeder when these two distances are the same.; i.e.:
∆ x s = ∆x p
Relative Velocity
.
Then,
∆xs = ∆xp → vst = ap ( t −1)
1
2
t 2 −15.89t +1 = 0 → t =
2
→
(13.89m s) t = 12 ( 2m s2 )( t 2 − 2t +1) = t 2 − 2t +1
15.89 ± 15.892 − 4
2
= 0.0632 s , 15.83 s
Since the police car doesn’t accelerate until t = 1.00 s , the correct answer is
t = 15.8 s
Force
We have introduced the concept of motion and obtained a set of equations that
describe the motion of an object given some initial conditions. We have also
introduced the concept of vector and have shown that velocity and acceleration are
vectors (have magnitude and direction).
But.. we have not discussed what set an object in motion, or change its state of
motion. This is what we will be learning in this chapter.
Force
The area that studies the connection between motion and its cause is called
dynamics.
The cause of the motion is force.
But what force is??
Force is the interaction between two objects.
For example: When you push a grocery cart which is initially at rest, you are exerting
a force on it.
The force you apply on the cart changes
its state of motion
Force
Force is the interaction between two objects.
Side note:
You may ask me: but how do two objects interact?
For example: You have seen that magnets attract some materials such as a piece of
iron. But the magnet does not need to touch the iron to attract it. So, how do these
two objects interact?
The answer to this question requires advanced physics. However, let’s say that there
are some special particles that operates as messengers between the two objects.
These messengers are the carries of the force between these objects and therefore
responsible for their interaction.
There are four fundamental kinds of forces:
Gravitational Responsible for having the Moon orbiting the Earth….
Electromagnetic Responsible for the attraction between the magnet and iron.
Weak Main responsible for the processes that make stars to shine
Strong Responsible for the nuclear interaction
Force
Note that the direction of the grocery cart depends on the direction of the force
you apply on it:
You can push it on a straight line, or
to the left, or to the right, etc..
The strength you push the cart will
determine how fast the cart goes or
how fast you will change its direction.
So, it is clear that you should assign not only
a direction, but also a magnitude to
the force you apply on the cart. In other
words,
Force is a vector.
And we will represent it using the standard vector representation:
with the arrow representing the direction a force is applied on an object.
Force
Forces being vectors can be added using the methods discussed in chapter 3.
Example: In the figure you and your mother are pushing the same grocery cart with
forces represented by
and
, respectively. The total force applied on
the cart,
, will be the vector addition of both forces:
You can then use, for example,
the method of components for
vector addition after having
selected an appropriate
reference frame and
coordinate system.
Force
In general we can state that if a system of two or more forces are applied on a object,
the net force on this object will be given by:
(4.1)
Where the subscript “ i ” in
denotes a particular force acting on the object where
“i “ can be any number between 1 and N, where N is the total number of forces being
applied on the object. The Greek symbol Σ denotes a summation over all possible
forces from 1 to N.
Example: Determine the net force
;
;
;
; and
acting on an object subject five different forces:
y
x
Newton’s Laws of Motion
Relationship between force and motion
We have seen that given a reference frame, an object can be found in any of the
following states of motion:
• at rest
• moving with constant velocity
• accelerated
But an applied force on an object can change its state of motion:
• The object can starting moving from rest
• Can change velocity from a state of constant velocity
• Can change direction (and therefore the vector velocity), and may be the
magnitude of its velocity.
It is clear that if NO forces are applied on an object, it will tend to remain on its
original state of motion.
Newton’s Laws of Motion
Newton’s First Law of Motion
We can generalize the previous statement to say that:
If the NET FORCE of all forces applied on a object is ZERO, then the object will tend
to remain in its state of rest, or constant velocity.
This follows from the fact that a non-zero net force will change the state of motion of
any object, which implies in changing its velocity either in direction or magnitude or
both. But, by definition, change in velocity implies in acceleration. So, if no
acceleration is present, the object will remain in its original state of motion.
We can then state the Newton’s first law of motion as:
“ Every Object continues in its state of rest, or uniform velocity in a straight
line, as long as no net force acts on it ”
Newton’s Laws of Motion
and
Example: An object initially at rest is suddenly subjected to two forces,
of same magnitude and opposite direction. What is the net force acting on this
object? Will it move?
Answer:
Let
But
be the net force, then:
, then
Since the net force is zero, the object will NOT move.
,
Newton’s Laws of Motion
Note 1:
You can always represent the forces on the previous diagram as
instead of
This can be generalized to any system of forces.
Newton’s Laws of Motion
Note 2:
Newton’s first law ONLY holds in inertial frames.
Inertial frames are frames FIXED on a system which is either at rest or moving
with constant velocity (no changes in direction and magnitude).
Or, in other words, inertial frames are those where the Newton’s first law is valid.
Frames other than the inertial frames are called non-inertial frames (for obvious
reasons).
An example of non-inertial frame is that of a frame fixed on an accelerated
system (example: a car).
Example: if you are in a car and fix the reference frame on it. You then decide to
accelerate it to increase its speed from a state of constant velocity. You will then
notice that anything free on the passenger sit will tend to move, though no net force
has been applied on them. If you crash your car on a wall, you will tend to move
forward even if no force has been applied on you.
Newton’s Laws of Motion
Example:
Assume you are in a truck and have fixed the reference frame on it. You then decide
to accelerate the truck to increase its speed from a state of constant velocity. You will
then notice that anything free on the passenger sit on anywhere on the truck will tend
to move, though no net force has been applied on them. If you crash your truck on a
wall, you will tend to move forward even if no force has been applied on you.
Now, for a person at rest relative to the ground, the box will see to continue moving
with constant velocity while the car is being accelerate (ignore friction effects).
Newton’s Laws of Motion
The Concept of Mass:
Mass measures that quantity of matter in a body (or object). It is given in units of Kg.
But then, you may ask me: What is matter????
The answer is even more complicate than that to understand how objects interact.
We will not discuss about it here.
A alternative definition can be given as:
Mass is a measure of the inertia of an object.
Inertia is the tendency an object has to continue in its state of motion.
You may have experienced the fact that it gets harder to move a box of some material
as you add more of the same material to the box. So you are adding mass. This
means that adding mass in the box makes it more difficult to change its state of
inertia. So, mass is related to inertia:
The larger is the mass of an object, the bigger is its inertia.
Newton’s Laws of Motion
Newton’s Laws of Motion
Newton’s Second Law of Motion:
We have seen that force and acceleration are related In fact, acceleration is the
product of an applied force.
We have also noticed that force and mass are also related more mass implies that
a stronger force is needed to change the state of motion of a body).
But how are these quantities related???
From the equation of motion we know that:
If you triple (or double, or etc) your acceleration a, you expect to triple the velocity
added to your motion, v - v0.
It is also observed that if you triple the applied force on an object, its change in
velocity will be triple.
It is then clear that force and acceleration are directly proportional. We say:
The symbol
means “proportional to”
Newton’s Laws of Motion
Newton’s Second Law of Motion:
It also has been observed that for the same applied force, the change in velocity will
depend on the mass of the object:
the greater the mass, the smaller is the change in velocity smaller is the
acceleration.
This, together with
let Newton to state his second law of motion
“ The acceleration of an object is directly proportional to the net force acting
on it, and is inversely proportional to its mass. The direction of the
acceleration is in the direction of the net force acting on the object ”
Newton’s Laws of Motion
Newton’s Second Law of Motion:
Newton’s second law can be summarized with the following equation:
where I have omitted the subscript “ i ” and the limits of summation: 1 and N (they are
implicitly assumed to be present).
This equation can be re-written to yield the well known relationship between force,
mass and acceleration:
(4.2)
Note that you can also write:
, where