Area Vector

Area of a given surface is generally regarded as a vector. This helps us to generalize many expressions. As well as change various relations into vector dot product form, which lends deeper insight into the respective phenomenon. The Area Vector of a plane surface has a magnitude equal to that of the given surface. Its direction is taken to be radially outward, perpendicular to the surface . For an open surface, one can choose any surface to be the outward direction (since there are two!!). However, for a closed surface, the outward direction is fixed.

 For example, if S represents a small elementary area of magnitude ∆S , then the area vector points normally from the surface. For the open surface, I can attribute two outward normals (both correct!). But once I have chosen a direction, I have to stick to that convention for the remainder of the surface and throughout my problem. The unit area vector is thus given as

ȧ = ∆S / |∆S|

The vector nature of area helps us in finding the projection of a given surface on another in a sweet way! It is the same as finding the shadow of a given surface on another. So if S is the given surface and suppose I want its projection on the xy plane. Now if this surface makes an angle θ with the xy plane, then the S area vector will make the same angle θ with the normal to the xy plane i.e. z axis. This we can see from purely geometrical considerations.  Now the shadow or projection of the area S ( lets call it S’) will have a magnitude given by Scosθ. We can recall that in vector theory projection of a vector is given by its dot product. Similar is the case here. Even for a surface, its projection is completely given by area vector dot product with the unit area vector of the desired direction.

S’ = S. ȧ’

Coordinate System

We use a right handed coordinate system. All our relations and expressions are derived and concluded based on the consistent use of the right handed coordinate system. This is the reason why students sometimes derive the wrong direction of magnetic force on a charged particle, because they might be using the wrong hand!!!  The x, y and z axes are so oriented that they form a right handed co ordinate system. It means that if I stretch the fingers of my right hand along the x axes ( with y axes perpendicular to my palm and thumb pointing upwards) , and turn my fingers towards y axis , the direction of my thumb will give the direction of positive z axis. If I did the same thing with my left hand, the direction of my thumb will give the negative z axis.

ixj=k                       jxk=i                       kxi=j

So by comparison we can properly affix vectors to axes in these formulas

v=ωxr                     τ=rxF    

F=i(LXB)                               F=q(vxB)

Reference Frames

Free Body Diagram

Free body diagrams are an important way to understand the behavior of a body in a certain environment. The laws of motion, the conservations laws etc help to analyze a problem, find out certain variables, certain unknowns. But before doing that, we need to have some data to work upon. Free body diagrams have quite an aesthetic way of laying out that data in front of us.

                In a free body diagram, we free a body from its environment while concentrating more on the interactions that body has with its environment.

 Thus, besides the body, we retain the forces (alongwith their directions) acting directly on the body due to its immediate environment. While dealing with translational motion it is better to replace the body by a point particle (which is the center of mass in general!). However, when faced with torques and rotation we need to take into account the proper points of application of the forces too.

While drawing the free body diagram we must always keep in mind Newton’s third law, which always holds well. It helps us to see forces which are not so readily apparent.

Direction of some forces may not be known beforehand. But one can still assume some direction and work thereon. Most importantly, one should notice that the forces should be the one acting directly on the body. Let’s take for example a body of mass m resting on another body of mass M. If we draw a free body diagram of m it should be something like this:

The forces acting directly on the body are mg due to earth’s gravitational pull and N, the normal contact force from mass M. Here, I would like to point out that by Newton’s Third Law, a normal reaction equal in magnitude to N, would act on M due to mass m. So the free body diagram of M looks something like this:

Phasors

An important tool in the analysis of any sinusoid quantity is Phasor and Phasor Diagram.

Any sinusoid has a angular frequency and phase difference.

Dealing with sinusoids becomes easier with the use of Phasors.

A Phasor is simply a rotating vector. It does more than just point in a direction- it rotates!

Simple Harmonic Motion, Sinusoidal Electric Current (Alternating Current) all can be studied with the help of Phasors. The length of the Phasor corresponds to the amplitude or maximum magnitude of the quantity. This quantity varies sinusoidally with time i.e. a function of any one the following forms

X = A sin (ωt + φ)

X = A cos (ωt + φ)

The instantaneous values of the quantity are show by the projection (shadow) of the Phasor on either of the x- or y- axis. Here, the Phasor rotates about the origin with an angular velocity ω and φ is the angle the vector makes with the x axis at the initial time(generally taken as t=0).