Kinematics

In physics, kinematics is the branch of mechanics concerned with the motions of objects without being concerned with the forces that cause the motion. In this latter respect it differs from dynamics, which is concerned with the forces that affect motion.

Coordinate systems

Fixed Rectangular Coordinates

In this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction from a non-rotating origin. Usually ec i , ! is a unit vector in the x direction, ec j , ! is a unit vector in the y direction, and ec k , ! is a unit vector in the z direction.

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The position vector, ec s , ! (or ec r , !), the velocity vector, ec v , !, and the acceleration vector, ec a , ! are expressed using rectangular coordinates in the following way:

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ec s = x ec i + y ec j + z ec k , !

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ec v = dot {s} = dot {x} ec {i} + dot {y} ec {j} + dot {z} ec {k} , !

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ec a = ddot {s} = ddot {x} ec {i} + ddot {y} ec {j} + ddot {z} ec {k} , !

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Note:

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dot {x} = rac{dx}{dt} , ddot {x} = rac{d^2x}{dt^2}

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Two Dimensional Rotating Coordinate Frame

This coordinate system only expresses planar motion.

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This system of coordinates is based on three orthogonal unit vectors: the vector ec i, and the vector ec j which form a basis for the plane in which the objects we are considering reside, and ec k about which rotation occurs. Unlike rectangular coordinates which are measured relative to an origin that is fixed and non rotating, the origin of these coordinates can rotate and translate - often following a particle on a body that is being studied.

Related Topics:
Orthogonal - Basis

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Derivatives of Unit Vectors

The position, velocity, and acceleration vectors of a given point can be expressed using these coordinate systems, but we have to be a bit more careful than we do with fixed frames of reference. Since the frame of reference is rotating, we must take the derivatives of the unit vectors into account when taking the derivative of any of these vectors. If the coordinate frame is rotating at a rate of ec omega , ! in the counterclockwise direction (that's omega ec k using the right hand rule) then the derivatives of the unit vectors are as follows:

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dot ec i = omega ec k imes ec i = omega ec j

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dot ec j = omega ec k imes ec j = - omega ec i

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Position, Velocity, and Acceleration

Given these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this coordinate system.

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Position

Position is straightforward:

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ec s = x ec i + y ec j

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It's just the distance from the origin in the direction of each of the unit vectors.

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Velocity

Velocity is the time derivative of position:

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ec v = rac{dec s}{dt} = rac{d (x ec i)}{dt} + rac{d (y ec j)}{dt}

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By the chain rule, this is:

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ec v = dot x ec i + x dot ec i + dot y ec j + y dot ec j

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Which from the identities above we know to be:

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ec v = dot x ec i + x omega ec j + dot y ec j - y omega ec i = (dot x - y omega) ec i + (dot y + x omega) ec j

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or equivalently

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ec v = (dot x ec i + dot y ec j) + (y dot ec j + x dot ec i) = ec v_{rel} + ec omega imes ec r

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where ec v_{rel} is the velocity of the particle relative to the coordinate system.

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Acceleration

Acceleration is the time derivative of position.

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We know that:

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ec a = rac{d ec v}{dt}

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