Recall: Kinematics

Displacement vector: Dr = rf - ri
Velocity vector: vavg = Dr/Dt
Instantaneous velocity: take the limit as Dt goes to zero.
Average acceleration: aavg = Dv/Dt
Example: P3-21
A car is on an incline that makes an angle of 24 degrees to the horizontal. The car rolls from rest down the incline with a constant acceleration of 4.00m/s2 for a distance of 50.0m to the edge of the cliff. The cliff is 30.0m above the ocean. Find (a) the car's position relative to the base of the cliff when the car lands in the ocean, and (b) the length of time the car is in the air.
  1. Reread the problem.
  2. Draw a sketch.
  3. Identify the data. For this problem, think in terms of (1) when the car is rolling down the incline, and (2) when the car is falling to the ocean.
  4. Choose the equations. For (1) vf=vi+at.
  5. Solve the equations.
  6. Check the answer.

Chapter 4: The Laws of Motion

We move from kinematics (description of motion) to dynamics (understanding the causes of motion, and the resulting effects). The causes of motion we call "forces". Some types of forces are familiar, like if you pull on a string or throw a ball. Other forces are just as real but less familiar, for instance, when you sit on a chair, the chair pushes upward, against your weight. Otherwise, you would fall through the floor!

Types of Forces

Forces can be classified as contact or field forces. Contact forces involve the contact of two objects, such as in the examples of someone pulling a string, throwing a ball, or sitting in a chair. Field forces are a bit more abstract. You may have heard the phrase "action at a distance" to describe gravity. Initially, most of the forces we will use are either contact forces or gravity.

Newton's Laws of Motion

The ideas embodied in Newton's three laws of motion were known to many. Newton gets credit primarily for sythesising what was then known into three laws, and demonstrating the power of his formulation by tackling the problem of planetary motion and gravitation.

Newton's First Law

What is the natural state of matter? Aristotle believed it is for matter to be at rest. But anybody who's driven on icy Michigan roadways can attest to the fallacy of that blief. Galileo proposed that the natural state is to resist acceleration, and this idea is formalized in the first law:
An object at rest remainst at rest, and an object in motion continues in motion with constant velocity (that is, constant speed in a straight line), unless it experiences a net external force.
Stated differently, if the sum of all external forces on an object is zero, then its acceleration is zero: SF=0 then a=0.
Example:
What are the forces acting on a person seated on a chair?

Newton's Second Law

The second law summarizes the observations about how an object accelerates when the net force is not zero:
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
We express this relation mathematically as: SF= ma. This is a vector equation, being satisfied for each component separately: SFx=max SFy=may SFz=maz
Units of Force and Mass
Recall that the SI unit of mass is the kilogram. Then, by dimensional analysis, we can see that the unit of force, call the newton, is 1N = 1kg m/s2. Although we often equate the unit of pound to kilograms, one is a weight (a force) while the other is a mass. The pound is related to the newton, 1N = 0.225lb. The weight of an object is due to the force of gravity on the mass of the object. In projectile motion, we saw that freely falling objects near the earth's surface experience an acceleration g, acting vertically downward. Since w = F = ma = mg, we see that the weight is mg. Because g varies little over the surface of the earth, it is easy to confuse weight and mass. In fact, we often measure the mass of objects by weighing them on spring scales. This difference may become more well understood as we enter the era of the space station.
Example: P4-4
A freight train has a mass of 1.5x107kg. If the locomotive can exert a constant pull of 7.5x105N, how long does it take to increase the speed of the train from rest to 80km/h?
  1. Reread the problem.
  2. Draw a diagram.
  3. Identify the data. vi=0, vf=80km/hr=22m/s, m=1.5x107kg, F = 7.5x105N.
  4. Choose equations. F=ma, and vf=vi+at.
  5. Solve equations. a=F/m, t=(vf-vi)/a= (vf-vi)m/F = (22m/s)(1.5x107kg)/7.5x105N = 440 kg m/sN = 440 s or about 7 and 1/3 minutes.
  6. Check the answer. Assuming that the values given are reasonable, 7 minutes is a reasonable result for a train.

Newton's Third Law

In nature, forces always come in pairs.
If two objects interact, the force exerted on object 1 by object 2 is equal in magnitude but opposite in direction th the force exerted on object 2 by object 1.
This law is commonly stated as "for every action, there is an equal and opposite reaction". This statement is fine, as long as you remember that action means the force exerted by 1 on 2, and reaction is the force exerted by 2 back on 1. Mathematically, we will write this as F1 on 2= -F2 on 1.