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Chapter 11: Problem 2

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

Short Answer

Expert verified
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

01

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.
02

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.
03

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

Differentiate the \(\hat{i}\) component of the position vector with respect to time: \(\frac{d}{dt}(r\cos(\omega t)\hat{i}) = -r\omega\sin(\omega t)\hat{i}\)
04

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

Differentiate the \(\hat{j}\) component of the position vector with respect to time: \(\frac{d}{dt}(r\sin(\omega t)\hat{j}) = r\omega\cos(\omega t)\hat{j}\)
05

Velocity Vector

Combine the derivatives for the \(\hat{i}\) and \(\hat{j}\) components: \(\vec{v}(t) = -r\omega\sin(\omega t)\hat{i} + r\omega\cos(\omega t)\hat{j}\)
06

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.

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