Elliptical motion about the origin

[Identify]

Homework Statement

A ball of mass m fastened to a long rubber band is spun around so that the ball follows an elliptical path about the origin given by:

r(t)=bcos(ωt)e(x)+2bsin(ωt)e(y)

b, ω constants
bold type indicates vectors

Find the period of the balls motion.

Homework Equations

r(t)=bcos(ωt)e(x)+2bsin(ωt)e(y)

The Attempt at a Solution

I think the period is, T=2Pi/ω because the motion is harmonic but I'm not sure if this applies for elliptical motion..?

 

[gneill]

Homework Statement

A ball of mass m fastened to a long rubber band is spun around so that the ball follows an elliptical path about the origin given by:

r(t)=bcos(ωt)e(x)+2bsin(ωt)e(y)

b, ω constants
bold type indicates vectors

Find the period of the balls motion.

Homework Equations

r(t)=bcos(ωt)e(x)+2bsin(ωt)e(y)

The Attempt at a Solution

I think the period is, T=2Pi/ω because the motion is harmonic but I'm not sure if this applies for elliptical motion..?

Hi Identify, Welcome to Physics Forums.

All you have to do is establish after what time period the function r(t) repeats. What do you know about finding the overall period of a function that is comprised of other functions with their own periods?

 

[Identify]

Thanks gneill.
I think when 2 periodic functions are added their periods are Pi(the lowest common multiple of the two periods). In this case the answer would be T=2Pi/w, since the period of the cos and sin functions are both 2Pi.

 

[gneill]

Thanks gneill.
I think when 2 periodic functions are added their periods are Pi(the lowest common multiple of the two periods). In this case the answer would be T=2Pi/w, since the period of the cos and sin functions are both 2Pi.

Your result is fine.

 

[Identify]

To find the distance from the origin I take,

|r(t)|=((bcos(ωt))^2 + (2bsin(ωt))^2))^1/2

Is there a way I can use the sin^2(u) + cos^2(u) = 1 identity to simplify this any further? Or is this the simplified form? If the identity can be used here I am having trouble with the b and 2b coefficients.

 

[gneill]

To find the distance from the origin I take,

|r(t)|=((bcos(ωt))^2 + (2bsin(ωt))^2))^1/2

Is there a way I can use the sin^2(u) + cos^2(u) = 1 identity to simplify this any further? Or is this the simplified form? If the identity can be used here I am having trouble with the b and 2b coefficients.

Well, you can pull the b out for starters. You could also convert either cos² or sin² via your identity to get everything in terms of just sin² or just cos².