https://file.aiquickdraw.com/mathbot/file/1728638246913ti7v4ll5.jpg

Let's analyze and solve the problem step by step.

### Problem Breakdown:

Given:
- The position vector of the particle \(\vec{r}\) is:
  \[
  \vec{r} = \cos(\omega t) \hat{i} + \sin(\omega t) \hat{j}
  \]
  where \(\omega\) is a constant.

We are asked to show:
1. (a) The velocity vector \(\vec{v}\) of the particle is perpendicular to \(\vec{r}\).
2. (b) \(\vec{r} \times \vec{v}\) is a constant vector.

---

### (a) Show that \(\vec{v}\) is perpendicular to \(\vec{r}\):

The velocity vector \(\vec{v}\) is the time derivative of the position vector \(\vec{r}\). 

Given:
\[
\vec{r} = \cos(\omega t) \hat{i} + \sin(\omega t) \hat{j}
\]

Find the velocity vector by differentiating \(\vec{r}\) with respect to \(t\):
\[
\vec{v} = \frac{d}{dt}\left[\cos(\omega t) \hat{i} + \sin(\omega t) \hat{j}\right]
\]

Applying the chain rule:
\[
\vec{v} = -\omega \sin(\omega t) \hat{i} + \omega \cos(\omega t) \hat{j}
\]

Now, for two vectors to be perpendicular, their dot product must be zero:
\[
\vec{r} \cdot \vec{v} = 0
\]

Compute the dot product:
\[
\vec{r} \cdot \vec{v} = \left[\cos(\omega t) \hat{i} + \sin(\omega t) \hat{j}\right] \cdot \left[-\omega \sin(\omega t) \hat{i} + \omega \cos(\omega t) \hat{j}\right]
\]

This expands to:
\[
\vec{r} \cdot \vec{v} = \cos(\omega t)(-\omega \sin(\omega t)) + \sin(\omega t)(\omega \cos(\omega t))
\]

Simplifying:
\[
\vec{r} \cdot \vec{v} = -\omega \cos(\omega t) \sin(\omega t) + \omega \sin(\omega t) \cos(\omega t)
\]
\[
\vec{r} \cdot \vec{v} = 0
\]

Thus, the velocity vector \(\vec{v}\) is perpendicular to the position vector \(\vec{r}\).

---

### (b) Show that \(\vec{r} \times \vec{v}\) is a constant vector:

The cross product of \(\vec{r}\) and \(\vec{v}\) is:
\[
\vec{r} \times \vec{v} = \left[\cos(\omega t) \hat{i} + \sin(\omega t) \hat{j}\right] \times \left[-\omega \sin(\omega t) \hat{i} + \omega \cos(\omega t) \hat{j}\right]
\]

Using the property of the cross product:
\[
\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = 0 \quad \text{and} \quad \hat{i} \times \hat{j} = \hat{k}
\]

Expanding the cross product:
\[
\vec{r} \times \vec{v} = \cos(\omega t)\omega \cos(\omega t)(\hat{i} \times \hat{j}) + \sin(\omega t)(-\omega \sin(\omega t))(\hat{j} \times \hat{i})
\]

Simplifying this:
\[
\vec{r} \times \vec{v} = \omega(\cos^2(\omega t) + \sin^2(\omega t))\hat{k}
\]

Using the trigonometric identity \(\cos^2(\omega t) + \sin^2(\omega t) = 1\):
\[
\vec{r} \times \vec{v} = \omega \hat{k}
\]

Thus, \(\vec{r} \times \vec{v} = \omega \hat{k}\), which is a constant vector (since \(\omega\) is constant and \(\hat{k}\) is a unit vector in the z-direction).

---

### Conclusion:
- (a) The velocity vector \(\vec{v}\) is perpendicular to the position vector \(\vec{r}\).
- (b) The cross product \(\vec{r} \times \vec{v}\) is a constant vector.

Would you like more details or have any questions?

Here are 5 related questions to further explore this topic:
1. What is the physical interpretation of a particle's velocity being perpendicular to its position vector?
2. How does this result relate to circular motion?
3. Can the result of \(\vec{r} \times \vec{v}\) being constant apply to non-uniform circular motion?
4. What happens if \(\omega\) is not a constant?
5. How does the magnitude of the cross product relate to angular momentum?

**Tip**: The cross product of position and velocity vectors is often related to angular momentum in physics.

A particle moves so that its position vector is given by r = cos(ωt)i + sin(ωt)j, where ω is a constant. Show that (a) The velocity vector of the particle v is perpendicular to r, (b) r × v is a constant vector.

This problem demonstrates how to show that the velocity vector of a particle is perpendicular to its position vector and that the cross product of position and velocity results in a constant vector. It involves differentiation and vector operations, suitable for university-level calculus and physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation