Question

What is the relationship between the position and velocity vectors for motion on a circle?

Short answer

Based on the solution provided above, briefly explain the relationship between position vectors and velocity vectors for motion on a circle. The relationship between the position and velocity vectors for motion on a circle is derived by differentiating the position vector with respect to time. The position vector (\(\vec{r}(t)\)) represents the location of a particle on a circle at time 't', while the velocity vector (\(\vec{v}(t)\)) represents the rate of change of position with respect to time. The components of the velocity vector are scaled by the radius and angular velocity of the circular motion.

Step-by-step solution

Position Vector for Motion on a Circle

For motion on a circle with radius 'r' and center at the origin, the position vector at any given time can be represented as: \(\vec{r}(t) = r*[\cos(\omega t)\hat{i} + \sin(\omega t)\hat{j}]\) where \(\omega\) is the angular velocity and 't' is the time.

Differentiation

To find the velocity vector, differentiate the position vector \(\vec{r}(t)\) with respect to time: \(\frac{d\vec{r}(t)}{dt} = \frac{d}{dt}(r*[\cos(\omega t)\hat{i} + \sin(\omega t)\hat{j}])\) Now we need to differentiate the two components separately.

Differentiate the \(\hat{i}\) component

Differentiate the \(\hat{i}\) component of the position vector with respect to time: \(\frac{d}{dt}(r\cos(\omega t)\hat{i}) = -r\omega\sin(\omega t)\hat{i}\)

Differentiate the \(\hat{j}\) component

Differentiate the \(\hat{j}\) component of the position vector with respect to time: \(\frac{d}{dt}(r\sin(\omega t)\hat{j}) = r\omega\cos(\omega t)\hat{j}\)

Velocity Vector

Combine the derivatives for the \(\hat{i}\) and \(\hat{j}\) components: \(\vec{v}(t) = -r\omega\sin(\omega t)\hat{i} + r\omega\cos(\omega t)\hat{j}\)

Relationship between Position and Velocity vectors

The relationship between the position vector \(\vec{r}(t)\) and the velocity vector \(\vec{v}(t)\) is: \(\vec{r}(t) = r*[\cos(\omega t)\hat{i} + \sin(\omega t)\hat{j}]\) \(\vec{v}(t) = -r\omega\sin(\omega t)\hat{i} + r\omega\cos(\omega t)\hat{j}\) The velocity vector is the derivative of the position vector with respect to time, and its components are scaled by the radius and angular velocity of the circular motion.