Newtonian Mechanics

Newton's First Law:

Therefore we can make a chart...

Constant velocity |
Changing velocity |

Velocity has constant magnitude and constant direction |
Velocity has changing magnitude, or changing direction, or both |

Acceleration = 0 |
Nonzero acceleration |

Net force = 0 |
Nonzero net force |

Everything in each column is a synonym for everything else in that same column.

Newton's Second Law:

Mass = m. Scalar. SI unit = kilogram (kg) (not the gram!). Mass measures how much matter there is in the object.

Force =
.
A force is a push or a pull on an object. Vector. Unit = Newton (N).
From Newton's Second Law, N = kg m / s^{2}.

Since force is a vector, we're not satisfied with Newton unless he can explain both its magnitude and its direction. From the Second Law we have:

net F = m a (the magnitude of the net force equals the mass times the magnitude of the acceleration). Intuition: : Increasing the net F (the size of the push) increases the acceleration (the response to the push). It's intuitive that increasing the amount of matter in the object makes it more difficult to alter its motion (it's harder to throw a bowling ball than a tennis ball), so it's intuitive that increasing mass should decrease the acceleration. For this reason we can think of the mass m as measuring the "inertia" of the object, where inertia is the difficulty in altering its motion.

(the direction of the net force is the same as the direction of the acceleration). Intuition: the object should accelerate in the direction it's being pushed.

Proof: Use Newton's Second Law to prove Newton's First Law. Proof here.

When solving mechanics problems:

1. Identify all the forces operating on the object.

2. Apply Newton's Second Law separately to each object; for each object, apply Newton's Second Law separately to the vertical and horizontal components (be sure to choose positive directions for each component).

Newton's Third Law:

Again, we expect Newton to describe both magnitude and direction:

F_{AB} = F_{BA} (the magnitude of the force of
object A on object B is equal to the magnitude of the force of object
B on object A)

And, the direction of the force of object A on object B is opposite to the direction of the force of object B on object A (this is the significance of the minus sign in the Third Law).

COMMON
QUESTION: If every force is balanced by an equal and opposite force,
doesn't that mean the net force is always 0? Answer: No. The key is
to apply the second tip from above--apply Newton's Second Law to each
object **separately**. For
example, suppose that objects 1 and 2 interact with each other and
with no other objects. Then
and
.
To summarize: every force on object 1 is paired with an equal and
opposite force **on a different object**,
so Newton's Third Law gives no particular reason for the net force **on
object 1** to be 0.

Types of Forces:

1. Gravitational Force

2. Electrostatic Force

3. Strong nuclear force

4. Weak nuclear force

All forces fall into one of these fundamental categories.

The Gravitational Force:

Gravitational forces come in pairs:

,
where F_{12} is the magnitude of the gravitational force of
object 1 on object 2, F_{21} is the magnitude of the
gravitational force of object 2 on object 1, G is the universal
gravitational constant, m_{1} and m_{2} are the
masses of the two objects, and r is the distance between the objects.

As for direction, the gravitational force is attractive. The direction of the gravitational force of object 1 on object 2 is toward object 1; the gravitational force of object 2 on object 1 points in the opposite direction, towards object 1.

You can see that the pair of gravitational forces satisfy Newton's Third Law.

The acceleration due to gravity. Consider an object in free fall near
the surface of the earth. Applying Newton's Second Law,
,
where m is the object's mass. Let's use the symbol
to
represent the acceleration due to gravity. Then
.
Direction of
must
be same as the direction of the gravitional force: down. As for
magnitude,
.
Substituting in for the net force using Newton's universal law of
gravitation,
,
where M is the mass of the earth and r is the distance between the
center of the earth and the center of the object. Cancelling m from
both sides,
.
If the object is close to the surface of the earth then the distance
between the center of the object and the center of the earth (r) is
approximately equal to the radius of the earth (R_{E}). Using
this approximation,
.
Notice that g does not depend on m; the acceleration of an object due
to gravity does not depend on the object's mass. Notice furthermore
that g will not change as the object falls. In sum, all objects in
free fall near the surface of the earth accelerate at the same,
constant rate. If you look up the universal gravitation constant (G),
the mass of the earth (M) and the radius of the earth (R_{E})
and plug them into the formula for g you will get g = 9.8 m/s^{2}.

Weight. The weight of an object is the gravitational force of the
earth on the object:
.
(More generally, the weight of an object is the gravitational force
of the nearest planet or satellite on that object.) Since it's a
force, weight is a vector. Since it's a force, the unit for weight is
the Newton. Using Newton's Second Law and Newton's Universal Law of
Gravitation,
,
i.e.,
,
where
is
the weight of the object, m is the object's mass, and
is,
as defined above, the acceleration due to gravity near the surface of
the earth (9.8 m/s^{2} down).

Magnitude. W = m g; the magnitude of an object's weight is the equal to the object's mass times the magnitude of the acceleration due to gravity.

Aristotle vs. Galileo. Aristotle thought bigger objects fell faster, but Galileo proved this was wrong (as long as the objects are big enough for air resistance to be negligible). All objects accelerate at the same rate. COMMON QUESTION: But don't more massive objects feel a bigger gravitational force? Doesn't Newton's Second Law say that a bigger gravitational force should translate into a bigger acceleration? Answer: Massive objects do indeed experience a bigger gravitational force (i.e., they weigh more), which tends to increase their acceleration. But more massive objects are also more difficult to accelerate, which tends to decrease their acceleration. The two effects just cancel. E.g., if you triple the mass you triple the force experienced; but you also triple the force necessary for any given acceleration. To summarize: From Newton's Second Law we know that , i.e., when the force is greater objects accelerate more, but objects with larger masses accelerate less; tripling the mass of an object triples both the gravitational force (numerator) and the mass itself (denominator) so overall the acceleration is the same.

*

Notice that the mass of an object determines two things--the gravitational force it experiences, and how the object responds to forces in general (its inertia). It is a interesting coincidence that doubling an object's inertia also doubles it's gravitational force; after all, doubling the inertia has no effect on, say, the electrostatic force.

*

The Normal Force.

Put a book on a table. Now, analyze the forces operating on the object. We know that it experiences its weight, which points down. But the book is not actually accelerating down. Therefore there must be another force on the book, a force pointing up. We refer to this as the normal force of the table on the book. A normal force can occur any time one object is in contact with another.

Thus, when object 1 is in contact with object 2, object 2 may exert a force called the normal force, , on object 1.

The direction of the normal force is perpendicular to the surface; e.g., the direction of the normal force on object 1 would be perpendicular to the surface of object 2.

The magnitude of the normal force, F_{N} can be determined
using Newton's Second Law.

Example: Suppose a 5 N book is resting on a table. What's the normal force on the book?

Solution: Identify all the forces operating on the book. There's the weight, pointing down. And there's the normal force, pointing up (perpendicular to the table). Let's choose up as the positive direction.

.
Therefore, in this case the magnitude of the normal force is equal to
5 N.

Example 2: Consider the same 5 N book resting on the table, but now I'm pushing down on it with 3 N force.

.
Now the magnitude of the normal force has increased to 8 N. It makes
sense that the normal force is bigger since it has to balance not
just the weight of the book but also my push.

Example 3. Now say I'm pulling up on the book with 3 N force.

Now
the normal force has a magnitude of only 2 N. It makes sense that the
normal force is smaller now since it's getting help from my own pull.
If I was pulling with 5 N, then the magnitude of the normal force
would be 0.

Notice from these examples that the magnitude of the normal force
is **not** always equal to the
object's weight. They are equal, though, if the normal force and the
weight are the only forces on the object, which often happens.

COMMON CONFUSION: Consider a book resting on a table. In the context of Newton's Third Law, what is the "reaction force" to the books weight? Students commonly think that the reaction force is the normal force, but this is wrong. Let's see why. Remember that the weight is the (gravitational) force of the earth on the book. Therefore the reaction force must be a force of the book on the earth. But the normal force is the force of the table on book. What is the correct reaction force to the book's weight, then? It is the gravitational force of the book on the earth. Remember that gravitational forces always come in action-reaction pairs.

The reason that it's tempting to say that the normal force is the reaction force is that the normal force, like the reaction force, does indeed have an opposite direction to the weight; and the normal force is often equal in magnitude to the weight--for example, when the weight and the normal force are the only 2 forces on the object, as in Example 1 above. But you can see from Examples 2 and 3 above that the normal force is not always equal in magnitude to the weight.

Recall that all forces fall into one of the four types of fundamental forces. So which type is the normal force? Electrostatic. The normal force between a table and a book resting on it arises from the electrostatic repulsions between the electrons in the book and the electrons in the table.

APPARENT WEIGHT

If you put an object on a scale, the scale will measure the force of the object on the scale, which we can call the apparent weight . If the scale is a rest, and we choose up as the positive direction, then

, and the apparent weight is equal (in magnitude) to the actual weight--this is why a scale is useful for determining an object's weight.

But now suppose that the object and scale are not at rest but are accelerating--say, because they're in an elevator. Then, , and the apparent weight is no longer equal to the true weight. Instead, we can see that if the elevator is accelerating up, the AW will be greater than the true weight; but if were accelerating down, the AW will be less than the true weight. (If the elevator is moving but not accelerating, the AW is the true weight.) Indeed, , so if the elevator has a downward acceleration of g, the apparent weight is 0.

So what if the elevator has a downward acceleration of more than g? Then the scale and object would fall to the ceiling of the elevator, which would seem as if it were the floor.

FRICTION

Kinetic friction. Kinetic friction occurs when one object is sliding along a surface; the kinetic friction is a force which the surface exerts on the object. We will use the symbol to stand for kinetic friction. Since its a force, kinetic friction is a vector; since its a force, the unit for kinetic friction is the Newton.

The direction of kinetic friction is to oppose sliding, parallel to the surface.

The magnitude of kinetic friction is , the surface's coefficient of kinetic friction times the magnitude of the normal force. Different surfaces have different coefficients of friction; a smooth surface like ice would have a low coefficient (resulting in low frictional forces); a rough surface like sandpaper would have a high coefficient (resulting in high frictional forces). When the magnitude of the normal force is large--i.e., when the object is pressing firmly against the surface--friction is large; when the normal force is small--i.e., when the object is pressing only lightly against the surface--friction is low; if the normal force is zero--if the object is not pressing against the surface at all--there's no friction.

Static friction. Tap a table lightly on the edge. The
table does not slide, even though you're exerting a force on
it. Therefore, the force of your tap must be counterbalanced by
some other force--what is this counterbalancing force? It
can't be kinetic friction, since we've already said that
kinetic friction only occurs when an object is sliding across a
surface, and the table isn't sliding. This shows that
there must be another type of friction than operates to *prevent*
sliding, which we can call static friction.

Static friction acts to prevent sliding. We will use the symbol to stand for kinetic friction. The direction of static friction, like kinetic friction, is to oppose sliding, parallel to the surface.

The magnitude of the static friction = exactly enough to prevent sliding.

Of course, there is some maximum static friction beyond which the
the object begins to slide and kinetic friction takes over. The
magnitude of the maximum static friction =
,
where
is
the coefficient of static friction, a constant particular to each
surface, and F_{N} is the magnitude of the normal force.

Notice then that there are two different formulas for the magnitude of the static friction:

actual = exactly enough to prevent sliding; or, whatever it takes to prevent sliding; but

maximum = .

Notice that the actual static friction has nothing to do with the normal force.

Whenever you attack a question about friction, then, you have to ask yourself two questions:

Am I dealing with kinetic friction or static friction?

If I'm dealing with static friction--do I want to find the actual static friction or the maximum static friction?

Recall again that all forces fall into one of the 4 fundamental types. Which type is friction? Electrostatic. Friction is caused by the repulsions between the electrons in the object and the electrons in the surface it is sliding along.

TENSION

Ropes exert forces on objects they are connected to, called "tension" forces. Here is an example of how to solve a problem involving ropes.

Example: Consider two objects, linked by a rope which winds around an axle as shown in the diagram below. Object 1 has mass 30 kg and rests on a table, Object 2 has mass 10 kg and hangs from the rope

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This page maintained by Steven Blatt. Suggestions, comments, questions, and corrections are welcome.