Constraint Equations for Particle on Rotating Ring

[GreenLRan]

Homework Statement

A particle of mass M is constrained to move on a ring of radius r which rotates about a vertical axis (Y) passing through the center at a constant angular speed (omega).

I am to find the constraint equation(s) for this system.

The origin of the system is at the bottom of the ring (so the maximum Y should be 2*radius)

Homework Equations

The Attempt at a Solution

So far, I have

X = r * cos(omega*t)
Z = r * sin(omega*t)

Where I am confused is how to get the height (Y) of the bead without specifying a new angle with respect to the X-Z plane. Can anyone help? Thanks!

 

[anathema]

Hello! I can help you with finding the constraint equations for this system. To find the height (Y) of the particle, we can use the Pythagorean theorem.

We know that the distance from the center of the ring to the particle is always equal to the radius (r). So, let's call this distance "d". Then, using the Pythagorean theorem, we can write:

d^2 = X^2 + Y^2

But, we also know that X = r * cos(omega*t) and Y = r * sin(omega*t), so we can substitute these values into the equation:

d^2 = (r * cos(omega*t))^2 + (r * sin(omega*t))^2

Simplifying this, we get:

d^2 = r^2 * (cos^2(omega*t) + sin^2(omega*t))

And since cos^2(x) + sin^2(x) = 1, we have:

d^2 = r^2

Taking the square root of both sides, we get:

d = r

Therefore, the height (Y) of the particle is always equal to the radius (r). So, the constraint equation for this system is:

Y = r

I hope this helps! Let me know if you have any further questions.