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Dynamics I
Written by YJ Soon ([email protected])
for the Raffles Institution Sec 3 GE Physics Programme
25 September 2006
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1. Newton’s First Law
Newton’s First Law, or the Law of
Inertia, states that an object tends to
continue in its state of rest or uniform
motion in a straight line unless acted
upon by an external force.
1+1=2
#
1
0
1" x 2 dx =
$
2
!
Mass and inertia
Figure 1: A giant ferret meets a nonsensically small killer
Inertia is the tendency of an object to
whale. Which has more inertia? Which is better at math?1
resist changes to its state of rest or state
of motion. Mass is a measure of the amount of inertia an object has – the more massive an
object, the more inertia it has.
Some examples
•
When you pull a tablecloth out from underneath a stack of plates, the plates tend to resist
change to their state of rest. Thus, when the tablecloth is pulled away, they simply fall under
the effect of gravity instead of being dragged along. If your pulling skillz are not so l33t,
then you end up with a lot of broken crockery and an upset parent. I meant this example
purely hypothetically, please don’t try at home.
•
When you’re sitting in a car and the driver suddenly turns left, your body tends to stay where
it was while the system “turns” left. This is why you feel as though your body is shifting
away from the direction of motion during a sudden turn.
•
When you throw an object in the air when an airplane starts accelerating from rest, the coin
tends to stay at the initial horizontal velocity it was at (zero), whereas everything around it
has started moving forward. Hence, it flies towards the back of the plane and slams into the
person sitting behind you, who will probably proceed to kick your seat for the next 12 hours
of your flight. Idiot.
1
The ferret for both. The whale has less mass and hence less inertia, and is off by a factor of 2 in his integral.
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2. Newton’s Second Law
Newton’s Second Law states that the acceleration of an object as produced by a net force is
directly proportional to the magnitude of the net force, in the same direction as the net
force, and inversely proportional to the mass of the object.
In unnecessarily large symbols, where F = force (in Newtons), m = mass (in kg), a =
acceleration (in m s-2), Newton’s Second Law is just:
F=ma
Figure 2: A giant equation.
Where force acts in a specified direction.
An alternative (out-of-syllabus) statement of Newton’s Second Law is that the rate of change of
momentum of a body is directly proportional to the resultant force acting on it and takes place in
the direction of the force. Momentum is a vector, the product of mass and velocity.
Zero resultant force
A direct implication of Newton’s Second Law is that when an object has zero resultant force
acting on it (e.g. being pushed to the left by 2 N and also being pushed to the right by 2 N), it
undergoes zero acceleration. This doesn’t mean the object is not moving, it just means the
object is not accelerating.
However, if there is a net (or resultant) force acting on an object, it will be undergoing a nonzero acceleration, in the direction of the net force. Here’s an illustrated example.
yawn
X
Y
Figure 3A: Equal and opposite forces are acting on the
chicken (X = Y). Resultant force is zero, hence the
chicken is either not moving or moving at constant
velocity.
wheee!
X
Y
Figure 3B: Forward force Y is greater than backward
force X. Hence, the chicken is accelerating forward.
Wheee!
Mass and Weight
Weight is the attractive force on an object due to gravity. Following the equation F = m a, we
can obtain the corresponding equation, W = m g, where W is weight (measured in Newtons), m
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is mass (measured in kg), g is acceleration due to gravity ( = 10 m s-2). Weight acts downwards,
or towards the centre of the Earth (see later section on “gravitational fields).
Example: Weight vs Force
In this example below, the aardvark has a mass of 20 kg, and is accelerating rightwards at 3.0 m
s-2. It’s a fat and fast aardvark.
N
F
W
Figure 4: An aardvark with a weight, and having a force act on it
The weight of the aardvark is:
W = m g = (20 kg) (10 m s-2) = 200 N
(Note that the unit Newton (N) is equivalent to kg m s-2 in SI units.)
The force F that the aardvark experiences forward, however, is:
F = m a = (20 kg) (3.0 m s-2) = 60 N
Note that even though both equations make use of the aardvark’s mass, they do so while acting
independently of each other. The aardvark is experiencing both forces, one accelerating him
forwards, as well as its own weight (which is countered by N, the upward reaction force from
the ground – see next section).
Example: Average airspeed velocity of an unladen swallow (not in syllabus)
Q: What is the average airspeed velocity of an unladen swallow?
A: What do you mean, an African or European swallow?
3. Newton’s Third Law
Newton’s Third Law states that when body A exerts a force on body B, body B will exert a
force on body A that is equal in magnitude and opposite in direction.
“For every action, there is an equal and opposite reaction” is a simpler description which
unfortunately doesn’t highlight the distinction that the respective forces are acting on different
bodies.
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Types of Forces
There are two types of forces we will consider:
•
•
Contact forces: Forces that act at a close range when one object touches another. E.g., when
you are pushed down the stairs, the person pushing you has exerted a contact force on you.
Forces at a distance: These are forces that act without contact. For example, gravity is a force
at a distance – the earth is pulling you even when you’re in the air. Magnetism and
electricity are other examples of forces at a distance.
Action-Reaction Pairs
For every action force, there is an equal and opposite reaction
force. These are called “action-reaction pairs.” Consider the
following example:
W = 10 N
A potato is resting on the table. What are the forces acting on
it?
First, there is W, the potato’s weight, or the force of gravity
on it. It has a magnitude of 10 N. Second, there is N, the
normal force of the table on the potato. It acts in an opposite
direction, and also has a magnitude of 10 N.
Note that W and N are not action-reaction pairs! W, the
weight, is a force at a distance exerted by the Earth on the
potato. Hence, its corresponding reaction force W’, is
actually the force the potato exerts on the Earth. This force is
of an equal magnitude and is opposite in direction.
W’ = 10 N
Figure 5: A staggeringly large
potato. Mmmmm.
The reaction pair of the normal force, N’, is simply the force the potato exerts on the table
because it is resting on it. This is a contact force.
4. Free-body diagrams
A free-body diagram is a very useful tool for figuring out what forces are acting in what
direction in a system. You’ll be required to know how to draw them on occasion, and in most
situations, drawing them out will help visualise the problem much more clearly than your
imagination can.
How to draw free-body diagrams
•
•
•
Identify the object(s) under consideration. If it’s just a single body accelerating across a
table, you’d probably just need to consider the body and not the table. If it’s a tractor pulling
a carriage, you’d want to include the tractor and the carriage and the cable in between.
Draw a simple shaded circle (not a square! – sorry if I misled you, but this is the preferred
convention) for each object in consideration. Label the object, optionally with its mass.
Draw in all forces acting on the object. Do not consider the reaction forces this object
exerts on other objects, nor any forces acting on other objects.
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•
•
•
•
•
Draw the forces as arrows leading away from the object. Hence, if an object is being pushed
to the right, draw an arrow from the object leading to the right, even though this seems to
imply that the object is being “pulled”.
When done, consider what the resultant direction of net force is. This could be different
from resultant direction of motion. Draw a double-arrow (“resultant arrow”) outside the
system.
Important note: This net force can be expressed as either of two things, the vector sum of
the forces in that direction, and also the Newton’s Second Law equivalent, i.e. mass !
acceleration. Equating these two is often an important step in obtaining your answer.
Not drooling all over your paper and smudging it is also often an important step in obtaining
your answer.
Sob with glee at your gloriously pretty free-body diagram.
Example free-body diagram
A 30 kg duck is accelerating across a table at 2.0 m s-2.
Fnet
R
F
Figure 6: Net force Fnet = F – R
OR Net force Fnet = mass ! acceleration = (30 kg)(2.0 m s-2) = 60 N.
Note the two ways the net force, Fnet, can be represented. If F = R, then Fnet = 0, and hence the
acceleration = 0 m s-2, hence the object is moving at constant velocity. Note that the vertical
velocities are balanced and can hence be ignored.
5. Example forces
Friction and drag forces
If you try to slide a body over a surface, the motion is resisted by a “bond” between the body
and the surface. This is called the frictional force, or just friction, and it acts along the surface
opposite to the direction of intended motion (e.g. R in figure 6 above).
Now, try these two experiments:
•
•
Push a book across a desk.
Push your desk across the room.
Obviously, the first object is easier to push. The book will slide for a while, and then come to a
stop because the table exerts a frictional force on the book and stops it. This is called kinetic
friction – “friction that opposes a moving object”.
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The second object, however, refuses to be pushed unless you push really hard and possibly hurt
your back (not my fault). This is because your applied force is balanced exactly by a static
frictional force (static friction), until a point where your applied force reaches a certain value,
called the limiting friction.
In general, friction acts to oppose motion, except in these two cases:
Figure 7A: Walking requires friction. As you step
backwards on the ground, friction prevents you from
slipping backwards and pushes you forwards. Imagine
walking on ice for what a frictionless walking surface
would feel like.
Figure 7B: When a car is being started, the wheel rolls
forward and friction allows it to “grip” the ground and
be pushed forwards. Thereafter, friction acts to
prevent motion and acts backwards. On a frictionless
surface, the car wouldn’t even start to move – the
wheels would just spin.
In all other cases, if you’re asked to figure out which direction friction is acting in (e.g. a car is
moving along, or a book is sliding across a table), it should be in a direction opposing motion.
Drag Forces
Other forces can conspire to oppose motion as well – e.g. air resistance if the object is moving in
air, or water resistance in water. This is how objects in free-fall stop accelerating and instead
reach terminal velocity. Let’s say the following politician is free-falling:
AAAAAAH
R
R
AAAAAAH
AAAAAAH
W
W
Fnet = W
Fnet = W - R
Figure 8A: When the politician
first starts falling, the only force
acting on him is weight W. Hence,
there is a net force downwards, and
he is accelerating downards.
Figure 8B: After falling for a bit,
the politician starts experiencing an
increasing air resistance R. His net
force downwards is hence reduced,
as is his acceleration.
Fnet = 0
W
Figure 8C: Finally, the air
resistance R has increased to the
point where it balances out W.
There is then no net force, no
acceleration and he is moving at
constant velocity towards certain
doom.
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Tension
When a cord (rope, string, intestine) is attached to a
body and pulled taut, the cord is under tension. It
hence pulls on the body with a force T in a direction
away from the body and along the cord.
Usually, the cord is considered to be light, so its
mass is negligible compared to that of the bodies,
and inelastic, so that it doesn’t stretch. It is hence just
a connector that pulls on both bodies with the same
magnitude T.
Some examples of tension are shown in the diagrams
shown to the right.
Figure 9: “The diagrams
shown to the right.”
6. Gravitational Field
This sub-chapter is simply an extension of everything you’ve seen so far in kinematics and
dynamics. You are expected to know that the mass of an object is the same whether on the Earth
or on any other planet, that the weight of an object is equal to its mass multiplied by the
acceleration due to gravity (or gravitational field strength), and that the Earth has a gravitational
field strength of 10 N/kg.
Measuring Mass and Weight
To measure weight, use the balance on the left below.
The object to be measured is hung off the spring
balance, and the extension of the spring is
proportional to the force exerted on it, i.e. Earth’s
gravitational pull on it, i.e. the weight. However, if
this experiment were to be repeated on the Moon, you
would obtain a different result for the object’s weight.
You would also probably be wasting a lot of
taxpayers’ money sitting around measuring the weight
of things on the Moon.
Figure 10: The last figure
To measure mass, use the balance on the right above. The object to be measured is placed in one
of the pans and its mass is compared to a number of standard masses in the other pan in order to
balance the two. This measurement of mass is valid whether on the Earth or on the Moon, and is
hence independent of the gravitational field strength.
END
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