**Uniform Circular Motion Analysis****Concept:**Consider a particle moving in uniform circular motion (UCM) in the counterclockwise direction along a circular path of radius \( r \). At the instant shown below, the position vector \( \vec{r} \) makes an angle \( \theta \) with the horizontal.**Diagram Description:**The diagram features a circle with radius \( r \). The position vector \( \vec{r} \) is drawn from the center to a point on the circumference, making an angle \( \theta \) with the positive x-axis. The coordinate axes (X and Y) intersect at the circle's center.**Tasks:**a) **Obtain the position vector in unit-vector notation.**b) **Obtain the velocity vector in unit-vector notation.**c) **Obtain the acceleration vector in unit-vector notation.**d) **Proof Tasks:** - Prove that the magnitude of the acceleration vector is \(\frac{v^2}{r}\) and it's direction is radially inward.e) **Prove that the position vector is perpendicular to the velocity vector.**

Answered: Image /qna-images/answer/12b1a1a3-166f-49f3-abf7-0e2f6ba92c03.jpg

3. Consider a particle moving in uniform circular motion (UCM) in the counter clockwise direction along a circular path of radius r as shown below. At the instant shown below the position vector r makes an angle 0 with the horizontal. 9. a) Obtain the position vector in unit-vector notation. b) Obtain the velocity vector in unit vector notation. c) Obtain the acceleration vector in unit vector notation. d) Prove that the magnitude of the radially inward. acceleration vector is direction is and it's e) Prove that the position vector is perpendicular to the velocity vector.

3. Consider a particle moving in uniform circular motion (UCM) in the counter clockwise direction along a circular path of radius r as shown below. At the instant shown below the position vector r makes an angle 0 with the horizontal. 9. a) Obtain the position vector in unit-vector notation. b) Obtain the velocity vector in unit vector notation. c) Obtain the acceleration vector in unit vector notation. d) Prove that the magnitude of the radially inward. acceleration vector is direction is and it's e) Prove that the position vector is perpendicular to the velocity vector.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

**Uniform Circular Motion Analysis**

**Concept:**
Consider a particle moving in uniform circular motion (UCM) in the counterclockwise direction along a circular path of radius \( r \). At the instant shown below, the position vector \( \vec{r} \) makes an angle \( \theta \) with the horizontal.

**Diagram Description:**
The diagram features a circle with radius \( r \). The position vector \( \vec{r} \) is drawn from the center to a point on the circumference, making an angle \( \theta \) with the positive x-axis. The coordinate axes (X and Y) intersect at the circle's center.

**Tasks:**

a) **Obtain the position vector in unit-vector notation.**

b) **Obtain the velocity vector in unit-vector notation.**

c) **Obtain the acceleration vector in unit-vector notation.**

d) **Proof Tasks:**
- Prove that the magnitude of the acceleration vector is \(\frac{v^2}{r}\) and it's direction is radially inward.

e) **Prove that the position vector is perpendicular to the velocity vector.**

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