Elliptical motion about the origin

In summary, the question asks to find the period of a ball's motion as it follows an elliptical path about the origin. The equation for the path is given by r(t)=bcos(ωt)e(x)+2bsin(ωt)e(y), where b and ω are constants. The period is found to be T=2Pi/ω, as the motion is harmonic and the period of both the cosine and sine functions is 2Pi. To find the distance from the origin, |r(t)|=((bcos(ωt))^2 + (2bsin(ωt))^2))^1/2, which can be simplified by pulling out the b coefficient and using the identity sin^2(u)
  • #1
Identify
12
0

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..?

 
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  • #2
Identify said:

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?
 
  • #3
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.
 
  • #4
Identify said:
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.
 
  • #5
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.
 
  • #6
Identify said:
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.
 

FAQ: Elliptical motion about the origin

What is elliptical motion about the origin?

Elliptical motion about the origin is a type of motion where an object moves in a curved path around a fixed point called the origin. The shape of the path is an ellipse, which is a stretched out circle.

What causes elliptical motion about the origin?

The primary cause of elliptical motion about the origin is the presence of a central force, such as gravity, that pulls the object towards the origin. This force is balanced by the object's inertia, which causes it to continue moving in a straight line. The combination of these two forces results in elliptical motion.

What are the key characteristics of elliptical motion about the origin?

The key characteristics of elliptical motion about the origin include a constant speed, changing direction, and a path that repeats itself over time. The object's motion is also governed by Kepler's laws of planetary motion, which describe the relationship between an object's distance from the origin and its speed.

How is elliptical motion about the origin different from circular motion?

The main difference between elliptical motion about the origin and circular motion is the shape of the path. In circular motion, the path is a perfect circle, whereas in elliptical motion, the path is an ellipse. Additionally, circular motion does not have a fixed origin, while elliptical motion is always about a fixed point.

How is elliptical motion about the origin used in real-world applications?

Elliptical motion about the origin is commonly seen in celestial bodies such as planets and moons orbiting around a central star or planet. It is also used in artificial satellites and spacecrafts that orbit around the Earth or other planets. In addition, elliptical motion is used in some types of machinery, such as steam engines, to convert linear motion into rotational motion.

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