Week 1: Newton’s Laws

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g)Fluid Forces, Pressure: A fluid in contact with a solid surface (or anything else) in general exerts a force on that surface that is related to the pressure of the fluid:

FP = P A

(32)

which you should read as “the force exerted by the fluid on the surface is the pressure in the fluid times the area of the surface”. If the pressure varies or the surface is curved one may have to use calculus to add up a total force. In general the direction of the force exerted is perpendicular to the surface. An object at rest in a fluid often has balanced forces due to pressure. The force arises from the molecules in the fluid literally bouncing o of the surface of the object, transferring momentum (and exerting an average force) as they do so. We will study this in some detail and will even derive a kinetic model for a gas that is in good agreement with real gases.

h) Drag Forces:

Fd = −bvn

(33)

(directed opposite to relative velocity of motion through fluid, n usually between 1 (low velocity) and 2 (high velocity). It arises in part because the surface of an object moving through a fluid is literally bouncing fluid particles o in the leading direction while moving away from particles in the trailing direction, so that there is a di erential pressure on the two surfaces, in part from “kinetic friction” that exerts a force component parallel to a surface in relative motion to the fluid. It is really pretty complicated – so complicated that we can only write down a specific, computable expression for it for very simple geometries and situations. Still, it is a very important and ubiquitous force and we’ll try to gain some appreciation for it along the way.

1.6: Force Balance – Static Equilibrium

Before we start using dynamics at all, let us consider what happens when all of the forces acting on an object balance. That is, there are several non-zero (vector) forces acting on an object, but those forces sum up to zero force. In this case, Newton’s First Law becomes very useful. It tells us that the object in question will remain at rest if it is initially at rest. We call this situation where the forces are all balanced static force equilibrium:

~	X	~	(34)
F tot =		F i = m~a = 0

i

This works both ways; if an object is at rest and stays that way, we can be certain that the forces acting on it balance!

We will spend some time later studying static equilibrium in general once we have learned about both forces and torques, but for the moment we will just consider a single example of what is after all a pretty simple idea. This will also serve as a short introduction to one of the forces listed above, Hooke’s Law for the force exerted by a spring on an attached mass.

Example 1.6.1: Spring and Mass in Static Force Equilibrium

Suppose we have a mass m stretched out some distance

hanging on a spring with spring constant k such that the spring is x from its unstretched length. This situation is pictured in figure 3.

We will learn how to really solve this as a dynamics problem later – indeed, we’ll spend an entire week on it! Right now we will just write down Newton’s laws for this problem so we can find a. Let the x direction be up. Then (using Hooke’s Law from the list above):