Hi Identify, Welcome to Physics Forums.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..?
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.
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.
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.
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.
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.
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.
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.