Understanding Uniform Circular Motion Calculations

[jenjen07]

Hi, I'm having trouble on a set of problems and I was wondering if someone could walk me through how to do one of these so I can at least attempt the rest. The problem says:

"A point P moves uniformly along the circle x[squared] + y[squared]= r[squared] with constant agular velocity w. Find the x and y coordinates of P at time t given that the motion starts at time t=0 and [theta] = [theta]initial. Then find the velocity and acceleration of the projection of P onto the x-axis and onto the y axis."

I found the derivative of the equation that was given but I have no idea where to go from there. Thank you for your help.

 

[arildno]

Welcome to PF!
How do you parametrize a circular trajectory?

 

[jenjen07]

what do you mean exactly? there's another little note on the problem that says the projection of p onto the x azis is the point (x,0) and the project of p onto the y-axis is the point (0,y)

 

[arildno]

Do you agree that since the particle moves in a circular orbit, we may write:
$$x(t)=r\cos(f(t)),y(t)=r\sin(f(t))$$
where f(t) is as yet undetermined.
With this choice, we are guaranteed that the particle moves on a circle, since:
$$x^{2}(t)+y^{2}(t)=r^{2}$$

 

[arildno]

If you agree so far, let's determine f(t)!
We know that the speed must be rw, since w is the constant angular velocity.
Differentiating to gain the velocity, we end up with the speed equation:
$$r|f'(t)|=rw$$
agreed? (|| signifies the absolute value)
This means f(t)=wt+K, where K is some constant.
If you have some particular questions to this, post them