Question

If I have given differential equation of motion and if I know a solution of that equation, how can I show that motion is planar?

Answer 1

if it contains 2 variables... it is planar

Answer 2

Equation of motion is this \[\ddot{r}=-ω^2r\] and its solution is \[\vec{r}=\vec{A}\cos(\omega t)+\vec{B}\sin(\omega t)\] where \[\vec{A},\vec{B}\] are constant vectors.

Answer 3

ω is constant, t is variable

Answer 4

it is equation of S.H.M (simple harmonic motion)

Answer 5

yes, but I have to show that it is planar

Answer 6

in SHM displacement of a particle can be represented as
y=a sinx , a is amplitude nd constant.... Therefore 2 variables are left.. hence planar

Answer 7

let's say that I don't know that. Is there a way to prove that motion is planar just using equations that I know, i.e. equation of motion, it's solution, which is (if I understood good) position vector and velocity?

Answer 8

I would like to know how to show it in general case, not in just this one.

Answer 9

put w=d^2r/dt ... try to integrate

Answer 10

\[dt\]or\[dt^2\]?

Answer 11

dt^2 coz acc is double derivative of displacement

Answer 12

This says it all :

\(\vec{r}=\vec{A}\cos(\omega t)+\vec{B}\sin(\omega t)\)

\(\vec{r}\) is a linear combination of \(\vec{A}\) and \(\vec{B}\), which define a plane. So the motion is planar.

Answer 13

@Vincent-Lyon.Fr what if A and B are three-dimensional vectors?

Answer 14

When \(\vec{OM}=\lambda \vec A + \mu \vec B\),
M is in a plane (or on a line if \(\vec A\) and \(\vec B\) are parallel)

\(\vec A\) and \(\vec B\) form a base of the plane.

Answer 15

if the solution is in 2 variable, then it's motion is planar.

Answer 16

in this solution, what are variables? I know that t is variable, but what is another one?

Answer 17

cos and sin are variables?

Answer 18

okay, when can you show that a motion is one dimensional?
only when there is one variable. variable means either the x, y or the z co-ordinate.
how do you plot a graph in a graph paper? you need a pair of variables . either the x-y or y-z or z-x. similarly, if the solution of your differential equation contains onnly two vaiables like the x-y or y-z or z-x, then you can say that the motion is planar.