Kinematics

From Academic Kids

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.

Because of its relative simplicity, kinematics is usually taught before dynamics or the concept of a force is introduced. The equations of motion are generally taught at secondary school level.

Contents

Fundamental equations

Relative motion

This is a simple equation from vector mathematics that restates vector addition: motion of A relative to O is equal to the motion of B relative to O plus the motion of A relative to B:

<math>r_{A/O} = r_{B/O} + r_{A/B} \,\!<math>

Rotating frame

One fundamental equation in kinematics is the equation for the derivative of a vector described in a rotating frame of reference. As a sentence, it is: the time derivative of a vector in a fixed frame is equal to the derivative of the vector relative to the rotating frame plus the cross product of the angular velocity of the frame with the vector. In equation form that is:

<math>\left.\frac{dr(t)}{dt}\right|_{X,Y,Z} = \left.\frac{dr(t)}{dt}\right|_{x,y,z} + \omega \times r(t)<math>

where:

r(t) is a vector

X,Y,Z is the fixed frame

x,y,z is the rotating frame

ω is the rate of rotation of the frame.

Coordinate systems

Rectangular coordinates

In this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction. Usually <math>\vec i \, \!<math> is a unit vector in the x direction, <math>\vec j \, \!<math> is a unit vector in the y direction, and <math>\vec k \, \!<math> is a unit vector in the z direction.

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

<math>\vec s = x \vec i + y \vec j + z \vec k \, \!<math>

<math>\vec v = \dot {s} = \dot {x} \vec {i} + \dot {y} \vec {j} + \dot {z} \vec {k} \, \! <math>

<math> \vec a = \ddot {s} = \ddot {x} \vec {i} + \ddot {y} \vec {j} + \ddot {z} \vec {k} \, \! <math>

Note: <math> \dot {x} = dx/dt \, \! <math> , <math> \ddot {x} = d^2x/dt^2 \, \! <math>

Normal and tangential coordinates

This coordinate system only expresses planar motion.

This system of coordinates is based on two orthogonal unit vectors, the vector <math> \vec e_t \, \! <math> that is tangential to the path of the particle, and the vector <math> \vec e_n \, \! <math> that is normal to the path of the particle. Unlike rectangular coordinates which are measured relative to an unmoving origin, these coordinates follow the particle along its path.

Kinematic constraints

A kinematic constraint is any condition relating properties of a dynamic system that must hold at all times. Below are some common examples:

Rolling without slipping

An object that rolls against a surface without slipping obeys the condition that the velocity of its center of mass is equal to the cross product of its angular velocity with a vector from the point of contact to the center of mass, or:

<math> v_G(t) = \omega \times r_{G/O} \,\!<math>

For the case of an object that does not tip or turn, this reduces to v = R ω .

Gears (no slip)

Similar to the case of rolling without slipping, this involves two bodies with the same motion at their contact point. For any bodies 1 and 2 the constraint is:

<math>r_1 \omega_1 = r_2 \omega_2 \,\!<math>

where

r is a radius

ω is an angular velocity

Inextensible cord

This is the case where bodies are connected by some cord that remains in tension and cannot change length. The constraint is that the sum of all components of the cord, however they are defined, is the total length, and the derivative of this sum is zero.

See example

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