Скачать или смотреть Find the position and velocity vectors for circular motion with increasing speed.

We find the position and velocity vectors for uniformly accelerated circular motion, given a constant angular acceleration of alpha, and zero initial conditions (initial angular position is zero and initial angular velocity is zero),.

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We are given a linearly increasing angular velocity omega=alpha*t, where alpha is constant, and told that both initial conditions are zero (position and angular velocity). In the first part of the problem, we are asked to find the angle as a function of time when angular velocity is increasing, and we do this by guessing the antiderivative and applying the initial conditions. This process was analogous to the derivation of the equations of motion in one dimension.

Next, we find the position vector for circular motion with increasing speed. Since the x component is given by rcos(theta) and the y component is given by rsin(theta), all we have to do is plug in our angle as a function of time and express the position vector in i-hat j-hat notation.

Next, we find the velocity vector for circular motion with angular acceleration. All we have to do is take the time derivative of the components, but we're careful to use the chain rule for the time derivative!

Finally, we use the dot product to show that the velocity and position vectors are always perpendicular. We take the dot product of velocity and position and show that it is always zero, independent of time, and this means velocity and position vectors are perpendicular.