Motion described in polar coordinates
DOWNLOAD
FREE
|
Acceleration in polar co-ordinates
Equations are omitted for technical reasons - download the original pdf
To determine the acceleration of an object in polar co-ordinates we must differentiate its velocity vector. [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] This means that the radical component of the acceleration is [Equation goes here - download the original to see it.] and the transverse component is [Equation goes here - download the original to see it.] Example A beautiful girl is sitting on a horse on a merry-go-round. She has dropped her purse and is foolishly trying to reach down to pick it up. The merry-go-round has a period of 10s. In order to prevent the girl from harming herself, Larry, the experienced operator is walking towards her along a radius of the merry-go-round. The radius of the merry-go-round is 8m. At a given time Larry is 4m from the centre and is travelling . If Larry has a mass of 75 kg find the force he has to exert in a radical direction in order to maintain his speed towards the damsel in distress. Find also the force he has to exert in a transverse direction in order to maintain his balance. Find the total force he has to exert in order to maintain his course of action. Solution [Equation goes here - download the original to see it.] The radical component of Larry's acceleration is [Equation goes here - download the original to see it.] Since [Equation goes here - download the original to see it.] The force Larry must exert in order to maintain his constant velocity towards the girl is [Equation goes here - download the original to see it.] The transverse component of Larry's acceleration is [Equation goes here - download the original to see it.] Here [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] Hence the force he must exert in the transverse direction is [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] The total force is: [Equation goes here - download the original to see it.] Example A particle is moving with constant angular velocity [Equation goes here - download the original to see it.]along a curve with equation [Equation goes here - download the original to see it.] Find the radical and transverse components of the acceleration in terms of . Find the acceleration of the particle. Solution We have [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] Now [Equation goes here - download the original to see it.] The radical component of the acceleration is [Equation goes here - download the original to see it.] The transverse component of the acceleration is [Equation goes here - download the original to see it.] The magnitude of the total acceleration is [Equation goes here - download the original to see it.] So the acceleration is constant.
|
Contents of
Motion described in polar coordinates
1 Motion described in polar coordinates. Vectors in polar co-ordinates
2 Change of basis
3 Rate of change of radial direction and radial angle
4 Velocity in polar co-ordinates
5 Acceleration in polar co-ordinates
|
