|
Chapter Four Lecture
 |
|
What are Forces?
|
|
|
 Formulas:
Demonstrations:
- Cart on Table
- Newton scale and weight
- Acceleration of Gravity on air track
Main Ideas:
- Newton's Three laws
- Weight
- Solving Problems
- Friction
 In this chapter we will study Newton's laws of motion. These are some of the
most fundamental and important principles in physics.
Isaac Newton (1642-1727)
 Newton was an English scientist and mathematician born into a poor farming family.
Luckily for humanity, Newton was not a good farmer, and was sent to Cambridge to study to become a preacher. At school, Newton studied mathematics, however, he was forced to leave
Cambridge when it was closed because of the Bubonic Plague. It was during this period that Newton made some of his most significant discoveries and developed some of his most profound theories in subjects such as
mathematics, physics, optics, alchemy, chemistry, and many other physical sciences. Newton did not, however, publish his results because he was particularly reticent to have
his ideas scrutinized by fellow scientists not because he feared they were in error, but because he did not want to spend the time defending his ideas to lesser minds (Newton had a bit of an ego problem).
However, Newton was not without his own problems suffering a mental breakdown in 1675 that lasted 4 years, but he never stopped working.
 Edmund Halley (of cometary fame) become interested in the orbits of astronomical objects
and finally convinced Newton to expand and publish his calculations and ideas. Newton devoted the period from August 1684 to spring 1686 to this task, and the result became one of
the most important and influential scientific works of all time, Philosophiae Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy" - 1687), often shortened
to Principia Mathematica or simply "The Principia.'' This work not only answered the questions about planetary orbits and the motions of astronomical objects, but it addressed the fundamentals of motion in
general. Newton's Laws of Motion set-up the structure upon which all motion is analyzed and described.
The key to understanding Newton's ideas about motion is to
investigate the description of "force" and "mass." The main idea is that "forces cause accelerations," and, that the amount of acceleration depends
on the "mass" of the object. Simply, a force is the reason why an object might accelerate.
|
|
Newton's First Law
|
|
|
Every object continues in its state of rest or of uniform speed in a straight line unless it is compelled to change that state by a "net force" acting on the object.
- This doesn't say that every moving object has a "force" acting on it.
- This doesn't say that a stationary object has no forces acting on it.
Newton's 1st Law says that if an object is stationary it will stay
stationary, and if it is moving, it will stay moving in a straight line, with constant speed, unless a net force acts on it. A net force means that the
sum of all the forces is not zero. A traffic light hanging above a road has a net force of zero. There is the force of gravity acting on the traffic light,
as well as tension forces in the wires from which the traffic light hangs. All of these forces are present and act on the traffic light, but the light does
not move. Objects that exert forces do not have to be moving.
Note also that a state of rest and a constant velocity are equivalent in
that both need a force to change them. A net force on an object is not needed to maintain the velocity, only to change velocity.
Push a book across a table. Why does it stop? [Friction] What if there
were no friction? If there were no friction, what would keep the book going? (Nothing, but it continues to move anyway! That's the point).
The force required to change a body's state of motion is a measure of the inertia of the body. Mass is a measure of the inertia of an object, where inertia is a descriptive term meaning an object's
resistance to a change of motion. Mass is simply a measure of the
quantity of matter in an object. It is harder to change the motion of something with more mass, whether or not there is air, gravity, etc. So
even in space where objects are weightless, it is harder to change the motion of a spaceship, than a wrench.
Therefore the amount of "acceleration" ("a") is inversely proportional to "mass" ("amount of material").
|
|
Newton's Second Law
|
|
|
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of the acceleration is in the direction of the applied net force.
More simply, a net force causes an acceleration. When a net force
exists, then, some change in the velocity of an object must occur (recall a change in velocity is an acceleration). If all the forces add up to zero -
then there is no net force and therefore no acceleration. If the forces do not add up to zero, then there is an acceleration in the direction of the net force.
The SI unit of force is
 . One Newton is the force required to accelerate one kg one meter per second per second. The British unit of force is the pound. (Note that pounds and kg are not a
measure of the same quantity). The pound is the unit of weight when the acceleration is
 . The unit of mass in the British system is
the slug, which is the mass that undergoes an acceleration of 1 ft/s2 when a force of 1 pound is applied to it. So where gravity is different, like on
the moon, the weight changes (pounds), but the mass stays the same (kg or slugs). (lbs = slugs ´ g). (1.0 lb º 4.48 N).
Note that the first law is a special case of the second. When SF = 0, a = 0, and so Dv = 0 because a = Dv/Dt.
|
 Problem: What is the average force exerted by a
shot putter on a 7.0 kg shot if the shot is moved through a distance of 2.8 m and is released with a speed of 13 m/s.
What is a? Use kinematical equations.
 |
|
|
|
Also note that the second law is a vector equation. You can break things down into components and say that SFy = may and SFx = max,
|
 Problem: Two forces, F1=45.0N and F2=25.0N act on a 5.00kg
block sitting on a table as shown. What is the horizontal acceleration (magnitude and direction) of the block?
F1x= F1cos(65.0) = 19.0 N
S F
x = max
19.0 N - 25.0 N = (5.00kg)ax
ax = -1.2 m/s2
What about in the y-direction?
F1sin(65.0) = 26 N. Is there an acceleration down? Why or why not?
|
|
|
|
|
|
Newton's Third Law
|
|
|
Whenever one object exerts a force on a second object, the second
object exerts an equal and opposite force on the first.
It is important to realize the equal and opposite forces are not exerted on the same body. The book gives some examples of this.
When I hammer a nail, the hammer exerts a force on the nail, which drives the nail into the wood, and the nail exerts an equal and opposite
force on the hammer which stops the hammer when it hits the nail. When determining if something will move (like will the nail go into the wood), we must always look only at the forces acting on that object.
|
 Problem: Suppose an
astronaut who is walking in space pushes on a spaceship with a force of 36N. The astronaut has a mass 92kg and the spaceship has a mass
of 11000kg. What happens?
So even though the forces are the same, the acceleration is not necessarily the same.
|
|
|
|
 Example: Suppose I throw a baseball. My body exerts a force on the baseball and the baseball
exerts a force back on my body. The baseball is accelerated. Is my body accelerated back? Let's look at all the forces on my body. The baseball pushes back on my body, and the ground pushes
forward with the same force as the baseball, so I do not accelerate. If the ground did not push back with the same force, I would accelerate. (Like standing
on ice). The Earth does accelerate backward, too, when there is friction.
Note the difference between Newton's Second Law and Newton's Third Law. Newton's Second law has to deal with all of the forces acting on a
single object. Newton's third law has to do with the forces acting and reacting on two different objects.
|
|
Force of Gravity & the Normal Force
|
|
|
 We know that an object near the surface of the earth will undergo an acceleration g =9.8 m/s2. Using Newton's second equation, we see
that for a body near the earth, the force of gravity is toward
F = mg
the center of the earth. This is what we call the object's weight.
Remember that the British unit "pounds" is a force, but the SI unit kg is a mass. So an object of 1 kg near the surface of the earth, has a "weight" of
F = (1kg)(9.8 m/s2) = 9.8 N which is 2.2 pounds.
On the moon where the gravity is 1/6 of the Earth's, the mass will still be 1 kg. Mass never changes no
matter what the gravity. However, the weight will now be
F = (1kg)(1.6 m/s2) = 1.6 N which is .37 pounds.
So the weight (which is a force) depends on the acceleration of gravity, but the mass is constant no
matter what the gravity, even if there is no gravity at all. When there is no gravity, it still takes more force to accelerate a spaceship of 11000 kg than to accelerate a man of 92 kg
at the same rate.("Inertia!")
 Now, suppose I put an object on a table. What are the forces acting on this object?
Certainly gravity is acting on it. But if there was only gravity, then the object would accelerate down by Newton's second law SF = ma. There must be a force pushing upon the book would accelerate down. This is the force of the table. It pushes up against the book.
It is equal and opposite to the force of the Earth pulling on the book. It is perpendicular to the table's surface so we call it the normal force. Normal
means perpendicular. This is not the equal and opposite force from Newton's Third Law. What is the equal and opposite force of the Earth pulling on the book? It is the book pulling on the Earth.
|
Problem: What is the normal force for a book with a mass of 0.80 kg?
Draw a free body diagram. SF = m a. What is the acceleration? It is 0, SF = 0. What are the forces? - Gravity and the normal force. What is the normal force?
Fn -mg = 0
Fn = mg = (0.80 kg)(9.8 m/s2 ) = 7.8 N.
|
|
|
|
|
Problem: Is the normal force always equal to the weight? NO. Suppose
I tie a string to the book and lift it with a force of 5.2 N. What is the normal force then?
We usually call the force exerted by a sting the "Tension." It is
always pointed in the direction of the string.
So now the normal is only 2.6 N. It is not always equal to the weight.
It is always equal to the amount needed to keep the object from accelerating downward.
|
|
|
|
|
Problem: If I push down with 5.2 N what is the normal force?
|
|
|
|
|
|
Solving Problems
|
|
|
Draw a diagram
- Consider only the forces acting on one object at a time and draw a free body diagram.
- Determine the components of each of the forces and set SF = ma in
each direction. (If there is no acceleration along a particular direction then SF = 0 along that direction.)
- Determine if there are other equations that need to be used (e.g. kinematic equations to get the acceleration).
- Solve the equations.
|
Problem: A man is pulling a wagon at a constant rate of 2.0 m/s with a
force of 52 N with the handle at an angle of 30°. What is the force of friction opposing the motion of the wagon?
If the wagon has a mass of 12 kg, what is the normal force?
 |
|
|
|
|
 Problem: A picture hangs from two
ropes as shown in the diagram. The tension in Rope 1 is 2.0´102N. What
is the tension in Rope 2 and the weight of the picture?
In x – direction:
In y - direction:
Therefore the mass of the picture can be found from;
 |
|
|
|
|
Problem: Frictionless table. What is the acceleration and tension in
the cord? The tensions must be the same and the accelerations must be the same, so for block 1 tension and acceleration are in the same direction.
Draw free body diagrams of both things, and look for what is the same.
Look at block falling down draw a free body:
For m1:
For m2:
We now have two equations that solve for the tension (T) so we can
set them equal to each other. In addition, if the string is not allowed to stretch then we also have that ax = ay
so I will just say that both blocks have the same acceleration "a."
 set equal to
 gives
(Not as fast as gravity alone, even though it is frictionless!)
Why? Inertia of block 1 is changing the acceleration!
 |
|
|
|
|
Problem: A train is given an acceleration of 5 m/s2. What is the tension
between the two cars with a mass of 2000 kg.
Free Body Diagram
Look at the forces on the middle car.
So if we knew Tension 1 we could solve the problem.
To get Tension 1, let's just look at the back car.
So to get the Tension 2, we plug this into the above equation.
 |
|
|
|
|
|
Friction & Inclines
|
|
|
In most cases we need to include friction. For many surfaces the force
of sliding friction is proportional to the normal force. That is, as the normal force increases, the force of friction increases linearly. When this is true, we write
 with the proportionality constant called mk, the coefficient of kinetic friction. This is the friction when the object is moving.
If I push a table and it doesn't move, there must also be a force
pushing back. That force comes from friction, as well. It is the friction when the object is not moving, or static, called the coefficient of static
friction. It is always greater than or equal to the coefficient of kinetic friction. It is often harder to start something sliding than to keep it sliding, due to the difference in these coefficients.
|
 Problem: A 20 kg sled is being
pulled across a horizontal surface at a constant velocity. The pulling force has a magnitude of 80.0 N and is directed at an angle or 30.0° above
the horizontal. Determine the coefficient of kinetic friction.
We know that
or
so to solve the problem, we need to find what Ffr and Fn are. Let's start
with Newton's second law in the vertical direction.
In the vertical y direction:
In the horizontal x direction:
 so
|
|
|
|
|
 Problem: Forces are being applied
to a box sitting on a surface with friction. Will the box move horizontally (along the surface)? F1=50N, F2=50N, Mass of the block 10kg, and ms=0.4.
Forces in x-direction:
The applied horizontal force is then;
So the combined applied forces are applying a force of 14.6N to the right.
How much force can static friction resist?
Forces in y-direction:
Force of Static Friction:
So will the box move? Since static friction can resist an applied force
up to 25N and the applied force is only 14.6N, the box will not move.
What force does F1 need to be before the box will move?
F1- F2x must be greater or equal to 25N
 therefore, F1³ 60.4N before the box will move to
the right. |
|
|
|
Many problems involving friction also involve surfaces at an incline. In
such a case it is almost always a good strategy to set one axis parallel to the surface and one axis perpendicular to the surface. Why? - Because
the acceleration is along the surface. Along the direction perpendicular to the plane, then SF=ma=0, and all of the acceleration is along the surface
parallel to the plane.
|
 Problem: An 18.0 kg box is released on a
37.0° incline and accelerates down the incline at 0.270 m/s2. Find the coefficient of friction and the frictional force. (Draw diagram)
Forces in y-direction are:
Forces in x-direction are:
or
Now, we can relate these to the coefficient of kinetic friction,
 |
|
|
|
|
|
Solving Some Hard Problems
|
|
|
|
Problem: Find the tension in each string. (skull's mass is 15.0 kg).
Look at free body diagram of skull.
Now look in the y-direction at the point where the three cables meet.
Along the y direction:
Along x-direction:
So we have 2 equations and 2 unknowns. Let's solve for T1 in the
first equation.
 plug the T1 result into the second equation.
Here we go!
And solve this for T2 (Looks like algebra fun!!).
This gives us the tension along T2. To get T1 we only have one step left.
This was a little harder because it required 3 equations and 3 unknowns.
|
|
|
|
|
 Problem: An electric motor is lowering a 452 kg crate with
an acceleration of 1.60 m/s2. What is the tension in the cord?
The only tricky part of this is to realize that all of the cords have the same tension, so
the free body diagram looks like this and (with tension in the opposite direction from the acceleration).
 |
|
|
|
|
Problem: Given the following situation with mk = 0.50, what is the acceleration of m1 and m2? (Since both objects move together
ay= ax = a.)
Look first at block that is hanging (m1):
Now for m2:
Now setting both T's equal and
 |
|
|
|
|
Problem: A 3.0 kg mass hangs at one end of a rope that is attached to a
support on a railroad car. When the car accelerates to the right, the rope makes an angle of 4.0° with the vertical. Find the acceleration of the car.
Look at the ball.
In the vertical direction:
In the horizontal direction:
 |
|
|
|
|
