Circular motion - determining expression for a position vector

[Linus Pauling]

1. Consider a particle--the small red block in the figure--that is constrained to move in a circle of radius R. We can specify its position solely by theta(t), the angle that the vector from the origin to the block makes with our chosen reference axis at time t. Following the standard conventions we measure theta(t) in the counterclockwise direction from the positive x axis.

Determine an expression for the position vector of a particle that starts on the positive y-axis at t = 0 (i.e., at t = 0, (x_0, y_0) = (0, R)) and subsequently moves with constant omega.
Express your answer in terms of R, omega, t, and unit vectors x_unit and y_unit.



2. r(t) = Rcos(omega*t)xhat + Rsin(omega*t)yhat



3. R*yhat + Rcos(omega*t)xhat

 

[Delphi51]

I may be confused - I see what looks like one question and two answers.
I don't see the answer to #1. When t = 0 your #2 answer works out to (R,0). And when t=0 your #3 answer works out to (R,R). The answer for #1 should work out to (0,R) when t = 0.

 

[kerimek]

+ Rsin(omega*t)yhat


Thank you for your question. I would like to clarify a few things before providing a response. First, it would be helpful to know what specific field or area of science this question pertains to, as the approach and terminology may differ. Additionally, the use of "x_unit" and "y_unit" in the expression is not clear. Are these referring to unit vectors in the x and y direction, respectively, or are they specific quantities with defined values?

Assuming that the expression is referring to unit vectors in the x and y direction, and based on the given information, we can approach this problem using vector mathematics. The position vector, r(t), of the particle at any time t can be expressed as the sum of its x and y components, where the x component is given by Rcos(omega*t) and the y component is given by Rsin(omega*t). Therefore, the position vector can be expressed as:

r(t) = Rcos(omega*t)x_unit + Rsin(omega*t)y_unit

Where x_unit and y_unit are unit vectors in the x and y direction, respectively. This expression takes into account the starting position of the particle at t = 0, which is on the positive y-axis at a distance of R from the origin. The constant omega represents the angular velocity of the particle, which determines its rate of change in position.

In summary, the expression for the position vector of a particle constrained to move in a circular motion can be determined by considering its x and y components and expressing them in terms of the given parameters R, omega, t, and unit vectors in the x and y direction. I hope this helps clarify and provide a satisfactory response to your question.