CLASSICAL MECHANICS PART 1 : CARTESSIAN , CYLINDRICAL, POLAR COORDINATES AND ROTATIONAL FRAME OF REFERANCE

 THE FOLLOWING CONTENT WILL COVER COMPLETE BASIC KNOWLEDGE ABOUT CLASSICAL MECHANICS AND GIVE YOU AN OVERVIEW ABOUT COORDINATE SYSTEMS AND FRAME OF REFERANCE......

Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

Cartesian coordinate system

The prototypical example of a coordinate system is the Cartesian coordinate system. In the plane, two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines.

In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes.^[5] This can be generalized to create n coordinates for any point in n-dimensional Euclidean space.

Depending on the direction and order of the coordinate axes, the three-dimensional system may be a right-handed or a left-handed system. This is one of many coordinate systems.

Polar coordinate system

Another common coordinate system for the plane is the polar coordinate system. A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r. For a given pair of coordinates (r, θ) there is a single point, but any point is represented by many pairs of coordinates. For example, (r, θ), (r, θ+2π) and (−r, θ+π) are all polar coordinates for the same point. The pole is represented by (0, θ) for any value of θ.

Cylindrical Coordinates

When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions.

In the cylindrical coordinate system, a point in space (Figure 11.7.1) is represented by the ordered triple (r,θ,z) where

• (r,θ) are the polar coordinates of the point’s projection in the xy-plane
• z is the usual z-coordinate in the Cartesian coordinate system

Conversion between Cylindrical and Cartesian Coordinates

The rectangular coordinates (x,y,z) and the cylindrical coordinates (r,θ,z) of a point are related as follows:

These equations are used to convert from cylindrical coordinates to rectangular coordinates.

• x=rcosθ
  
• y=rsinθ
  
• z=z

These equations are used to convert from rectangular coordinates to cylindrical coordinates

1. r2=x2+y2
   
2. tanθ=yx
   
3. z=z

Spherical Coordinates

In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances (r and z) and an angle measure (θ). In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates.

Definition: spherical coordinate system

In the spherical coordinate system, a point P in space (Figure 11.7.9) is represented by the ordered triple (ρ,θ,φ) where

• ρ (the Greek letter rho) is the distance between P$P$ and the origin (ρ≠0);
  
• θ is the same angle used to describe the location in cylindrical coordinates;
• φ (the Greek letter phi) is the angle formed by the positive z-axis and line segment OP¯, where O is the origin and 0≤φ≤π.

Converting among Spherical, Cylindrical, and Rectangular Coordinates

Rectangular coordinates (x,y,z)$(x,y,z)$, cylindrical coordinates (r,θ,z),$(r,\theta ,z),$ and spherical coordinates (ρ,θ,φ)$(\rho ,\theta ,\phi )$ of a point are related as follows:

Convert from spherical coordinates to rectangular coordinates

These equations are used to convert from spherical coordinates to rectangular coordinates.

• x=ρsinφcosθ
  
• y=ρsinφsinθ
  
• z=ρcosφ
  

Convert from rectangular coordinates to spherical coordinates

These equations are used to convert from rectangular coordinates to spherical coordinates.

• ρ2=x2+y2+z2
  
• tanθ=yx
  
• φ=arccos(zx2+y2+z2−−−−−−−−−−√).x

Convert from spherical coordinates to cylindrical coordinates

These equations are used to convert from spherical coordinates to cylindrical coordinates.

• r=ρsinφ
  
• θ=θ
  
• z=ρcosφ
  

Convert from cylindrical coordinates to spherical coordinates

These equations are used to convert from cylindrical coordinates to spherical coordinates.

• ρ=r2+z2−−−−−−√
  
• θ=θ
  
• φ=arccos(zr2+z2−−−−−−√)

Frame of reference

In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame

For n dimensions, n + 1 reference points are sufficient to fully define a reference frame. Using rectangular (Cartesian) coordinates, a reference frame may be defined with a reference point at the origin and a reference point at one unit distance along each of the n coordinate axes

In Einsteinian relativity, reference frames are used to specify the relationship between a moving observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes "observational frame of reference" (or "observational reference frame"), which implies that the observer is at rest in the frame, although not necessarily located at its origin. A relativistic reference frame includes (or implies) the coordinate time, which does not equate across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent.

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Non-inertial frames

Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.

An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x′, y′, a′.

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r′. From the geometry of the situation, we get

${\displaystyle \mathbf {r} =\mathbf {R} +\mathbf {r} '.}$

Taking the first and second derivatives of this with respect to time, we obtain

${\displaystyle \mathbf {v} =\mathbf {V} +\mathbf {v} ',}$

${\displaystyle \mathbf {a} =\mathbf {A} +\mathbf {a} '.}$

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.

These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as

${\displaystyle \mathbf {F} =m\mathbf {a} =m\mathbf {A} +m\mathbf {a} '.}$

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect).

A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation (see Fictitious force for a derivation):

${\displaystyle \mathbf {a} =\mathbf {a} '+{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '+2{\boldsymbol {\omega }}\times \mathbf {v} '+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')+\mathbf {A} _{0},}$

or, to solve for the acceleration in the accelerated frame,

${\displaystyle \mathbf {a} '=\mathbf {a} -{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '-2{\boldsymbol {\omega }}\times \mathbf {v} '-{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')-\mathbf {A} _{0}.}$

Multiplying through by the mass m gives

${\displaystyle \mathbf {F} '=\mathbf {F} _{\mathrm {physical} }+\mathbf {F} '_{\mathrm {Euler} }+\mathbf {F} '_{\mathrm {Coriolis} }+\mathbf {F} '_{\mathrm {centripetal} }-m\mathbf {A} _{0},}$

where

${\displaystyle \mathbf {F} '_{\mathrm {Euler} }=-m{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '}$ (Euler force),

${\displaystyle \mathbf {F} '_{\mathrm {Coriolis} }=-2m{\boldsymbol {\omega }}\times \mathbf {v} '}$ (Coriolis force),

${\displaystyle \mathbf {F} '_{\mathrm {centrifugal} }=-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')=m(\omega ^{2}\mathbf {r} '-({\boldsymbol {\omega }}\cdot \mathbf {r} '){\boldsymbol {\omega }})}$

Rotating reference frame

A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.

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