In the preceding chapter, we studied kinematics, which is the description of motion in terms of an object's position, velocity, and acceleration. In this chapter, we'll begin our study of dynamics, which is the explanation of motion in terms of the forces that act on an object.
Simply put, a force is a .push or pull exerted by one object on another. If you pull on a rope attached to a crate, you create a tension in the rope that pulls the crate. When a sky diver is falling through the air~ the Earth exerts a downward pull called the gravitational force, and the air exerts an upward force called air resistance. When you stand on the floor, the floor provides an upward, supporting force called the normal force. If you slide a book across a table, the table exerts a frictional force against the book, so the book slows down and eventually stops. Static cling provides a simple
example of the electrostatic force. (In fact, all of the forces mentioned above-with the exception of
gravity~are due ultimately to the electromagnetic force.)
How forces affect motion is described by three physical laws, known as Newton's laws. They form the foundation of mechanics, and you should memorize them.
NeWToN's FIRST LAw
An object's state of motion-its velocity-will not change unless a force acts on the object.
That is, if no net force acts on an object, then:
if the object is at rest, it will remain at rest;
or
if the object is moving, then it will continue to move with constant velocity (constant speed in a straight line).
The First Law says that objects naturally resist changing their velocity. In other words, objects at rest don't just suddenly start moving all on their own. Some external source must exert a force to make them move. Also, an object that's already moving doesn't change its velocity either-.-it doesn't go faster, or slower, or change direction all by itself; something must exert some force on it to make any of these changes happen. This property of objects, their natural resistance to change in their state of motion, is called inertia. In fact, the First Law is often referred to as the Law of Inertia.
The mass of an object is the quantitative measure of its inertia; intuitively, mass measures how much matter is contained in an object. Two identical boxes, one empty and one full, have different masses. The full box has the greater mass, because it's full: it contains more stuff, and more stuff means more mass. Mass is measured in kilograms, abbreviated kg. (Note: An object whose mass is 1 kg weighs a little more than 2 pounds on Earth, but be careful not to confuse mass with weight; they're different things.) Compared to an object whose mass is just 1 kg, an object whose mass is 100 kg has 100 times the inertia. Intuitively, we'd find it 100 times more difficult to cause the same change in its motion than that of the 1 kg object. This point will be clearer after we state the second of Newton's laws.
32 MCAT PHYSICAL SCIENCES REVIEW
NEWTON'S SECOND LAW
IfF net is the net-or total-force acting on an object of mass m, then_ the resulting acceleration of the object, a, satisfies this simple equation:
Fnet= ma
Forces are represented by vectors, because a force has a magnitude and a direction. If two different forces-let's call them F1 and F2-act on an object, then the total-or net-force on the object is simply the sum of these individual forces: Fnet = F1 + F2• Since forces are vectors, they must be added as vectors; that is, their directions must be taken into account. The following figures show some examples of obtaining F net from the individual forces that act on an object:
~F1
~Fnet n
F1 • D
n
o--.-Fnet
Note the following facts about the equation F net = ma:
F2 ..
F2t o---F1
~~
1) F net is the sum of all the forces that act on the object-namely, the object whose mass, m, is on the other side of the equation. Any force exerted by the object is not included in F netã
2) Because m is a positive number, the direction of a is always the same as the direction of F netã
Therefore, an object will accelerate in the direction of the net force it feels. This does not mean that an object will always move in the direction ofF netã Be sure that this distinction makes sense, because it can be a source of confusion-and therefore a potential MCAT trap. The direction of an object's velocity is not given by Newton's Second law; the direction of an object's acceleration is.
3) What ifF net= 0? Then a= 0. What does a= 0 mean? It means that the object's velocity does not change, which is also what Newton's First Law says. How about this question: Does F net= 0 mean that v = 0? Not necessarily! F net= 0 means that an object won't accelerate-not that it won't move-so if it's already moving at, say, 100 m/s toward the north, then it will continue to move at 100 m/s toward the north as long as the net force on the object remains zero. Be sure that you understand why F net = 0 does not necessarily mean that v = 0, because this is another potential MCAT trap.
4) Since F net = ma is a vector equation, it automatically means that the components of both sides must be the same. In other words, F net could be written as the sum of a force in the horizontal direction, (F net~x) plus a force in the vertical direction (F net,l); these would be the horizontal and vertical components ofF netã The equation F net = ma would then tel us that
Fnet,x =max and Fnet,y =may
So, dividing the horizontal component of the net force by m gives us the horizontal component of the object's acceleration, and dividing the vertical component of the net force by m gives us the vertical component of the object's acceleration.
5) The unit of force is equal to the unit of mass times the unit of acceleration:
[f] = [m][a] =kgã m/s2
A force of 1 kgãm/s2 is called 1 newton (abbreviated N). On Earth, a force of 1 N is about equal to a quarter of a pound, or about the weight of a medium-sized apple.
MCAT PHYSICS - CHAPTER 2: MECHANICS I 33
NewToN's THIRD LAw
If Object 1 exerts a force F1_0n_2 on Object 2, then Object 2 exerts a force, F2-on-l' on Object 1.
These forces, F1_0n_2 and F2_0n_1, have the same magnitude but act in opposite directions, so F 1-on-2 =-F 2-on-1
and they act on different objects. These two forces are said to form an action-reaction pair.
~ F2-on-1 F
This is the law commonly remembered as, "To every action, there is an equal but opposite reaction."
Unfortunately, this popular version of Newton's Third law can lead to confusion. All Newton's Third law says is that the forces in an action-reaction pair have the same magnitude and act in opposite directions (and on "opposite" objects). It does not say that the effects of these forces will be the same.
For example, suppose that two skaters are next to and facing each other on a skating rink. Let's say that Skater 1 has a mass of 50 kg and Skater 2 has a mass of 100 kg. Now, what if Skater 1 pushes on Skater 2 with a force of 50 N? Then F1-on-2 =50 Nand F2_on-l =-50 N, by Newton's Third law.
100 kg
But will the effects of these equal-strength forces be the same? No, because the masses of the objects are different. The accelerations of the skaters will be
a = F2-on-1 =-50 N = -1.!!1
1 m1 50 kg s2 and a F,
2 = 1-on-2 _ 50 N
m2 -100 kg =0.5:;
So, Skater 2 will move away with an acceleration of 0.5 m/ s2, while Skater 1 moves away, in the opposite direction, with an acceleration of twice that magnitude, 1 m/s2•
81 =-1 m/s2
2 82 = 0.5 rnls2
Therefore, while the forces are the same (in magnitude), the effects of these forces-that is, the resulting accelerations( and velocities)-are not the same, because the masses of the objects are different.
Newton's Third law says nothing about mass; it only tells us that the action and reaction forces will have the same magnitude. So, the point is not to interpret l i equal but opposite reaction" as meaning
"equal but opposite effect," because if the masses of the interacting objects are not the same, then the resulting accelerations (and velocities) of the objects will not be the same.
34 MCAT PHYSICAL SCIENCES REVIEW
~Example 2ã1: An object of mass 50 kg moves with a constant velocity of magnitude 1000 m/s. What is the net force on this object?
Solution. If the object moves with constant velocity, then the net force it feels must be zero, regardless of the object's mass or speed .
...._Example 2ã2: The net force on an object of mass 10 kg is zero. What can you say about the speed of this object?
Solution. If the net force on an object is zero, all we can say is that it will not accelerate; its velocity may be zero, or it may not. Without more information, we cannot determine the object's speed; all we know is that whatever the speed is, it will remain constant.
~Example 2ã3: For 6 seconds, you push a 120 kg crate along a frictionless horizontal surface with a constant force of 60 N parallel to the surface. If the crate was initially at rest, what will its velocity be at the end of this 6-second time interval?
Solution. Using Newton's Second law, we find that the acceleration of the crate is a= F lm = (60 N)/(120 kg)= 0.5 m/s2• Using Big Five #2, we now find that
v = v0 +at= 0 + (0.5 m/s2)(6 s) = 3 m/s.
~Example 2-4: For 6 seconds, you pull a 120 kg crate along a frictionless horizontal surface with a constant force of 60 N directed at an angle of 60° to the surface. If the crate was initially at rest, what will its horizontal velocity be at the end of this 6-second time interval?
Solution. To fmd the horizontal velocity, we need the horizontal acceleration.
Using Newton's Second law, we find that the horizontal acceleration of the crate is ax= Fxlm = (Fx cos 8)/m = (60 N cos 60°)/(120 kg)= (30 N)/(120 kg)= 0.25 m/s2•
Using Big Five #2, we now find that vx = v0x + ai = 0 + (0.25 m/s2)(6 s) = 1.5 m/s.
MCAT PHYSICS - CHAPTER 2: MECHANICS I 35 .,_Example 2-5: Two crates are moving along a frictionless horizontal surface. The
first crate, of mass M = 100 kg, is being pushed by a force of 300 N.
The first crate is in contact with a second crate, of mass m =50 kg.
SolutiQn.
F .... 1 M
(a) What's the acceleration of the crates?
(b) What's the force exerted by the larger crate on the smaller one?
(c) What's the force exerted by the smaller crate on the larger one?
(a) The force F is pushing on a combined mass of 100 +50= 150 kg, so by Newton's Second law, the acceleration of both crates will be a = (300 N) I (150 kg) = 2 m/ s2• (b) Because M and m are in direct contact, each is pushing on the other with a certain force. Let F2 be the force that M exerts on m. Then we must have F2 = ma, so F 2 = (50 kg)(2 m/ s2) = 100 N.
(c) By Newton's Third law, if the force that M exerts on m is F2, then the force that m exerts on M must be -F2• So, if we call"to the right" our positive direction, then the force that m exerts on M is -100 N. We can check that this is correct by looking at all the forces acting on M. We have F pushing to the right and -F2 pushing to the left.
The net force onM is therefore FnetonM = F + (-F2) ~ (300 N) + (-100 N) = 200 N. If this is correct, then FnetonM should equal Ma. Since M = 100 kg and a= 2 m/s2, we get Ma = 200 N, which does match what we found for FnetonM' (In effect what's happening here is that M is using 200 N of the 300 N force from F for its own motion and passing the remaining 100 N along tom, so that both move together with the same acceleration.)
.,_ Example 2-6: Two forces act on an object of ma.Ss m = 5 kg. One of the forces has a magnitude of 6 N, and the other force, perpendicular to the first, has a magnitude of 8 N. What's the acceleration of the object?
Solution. Forces are vectors, and when we find the net force on this object, we see that it's the hypotenuse of a 6-8-10 right triangle.
m
Since Fnet = 10 N, the acceleration of the object will be a= Fne/m = (10 N)/(5 kg)=
2 m/s2•
36 MCAT PHYSICAL SCIENCES REVIEW
.,.. Example 2-7: The figure below shows all the forces acting on a 5 kg object.
The magnitude of F1 is 50 N. If the acceleration of the object is 8 m/s2, what's the magnitude of F2?
F2~F1
Solution. The net force on the block is just the sum of F1 and F2, so F net= F1 + F2 = (+50 N) + F2, if we call "to the right" our positive direction. The net force must be ma = (5 kg)(8 m/s2) = 40 N. Since (+50 N) + F2 must be 40 N~ we.know that F2 = -10 N; that is, F2 has magnitude 10 N (and points to the left) .
.,.. Example 2-8: According to Newton's Third law, every force is "accompanied by"
an equal but opposite force. If this is true, shouldn't these forces cancel out to zero? How could we ever accelerate an object?
Solution. The answer does not involve the masses of the objects; Newton's Third law says nothing about mass. The key is to remember what F net means; it's the sum of all the forces that act on an object, not by the object. Let's say we have a pair of objects, 1 and 2, and an action-reaction pair of forces between them, and we wanted to find the acceleration of Object 2. We'd find all the forces that act on Object 2. One of these forces is F1_0n_2 • The reaction force,.F2_on-t' is not included in F net-on-2 because it doesn't act on Object 2; it's a force by Object 2. So, the reason why the two forces in an action-reaction pair don't cancel each other is that we'd never add them in the first place, since they don't act on the sameã object.