A probe caught in a tornado is moving in a circular path in the horizontal plane with approximately constant speed. You have three observations of the position of the probe: $$ \begin{array}{ll} \hline t & \mathbf{r} \\ \hline 0.0 \mathrm{~s} & 35.7 \mathrm{~m} \mathbf{i}+35.6 \mathrm{~m} \mathbf{j} \\ 1.0 \mathrm{~s} & 12.2 \mathrm{~m} \mathbf{i}+49.3 \mathrm{~m} \mathbf{j} \\ 2.0 \mathrm{~s} & -14.6 \mathrm{~m} \mathbf{i}+44.9 \mathrm{~m} \mathbf{j} \\ \hline \end{array} $$ (a) Find the average acceleration of the probe. (b) Find the center of the circle and the radius of the circle. You can use approximations as you see fit. (c) Find an expression for the position of the probe as a function of time.

Step by step solution

01

- Calculate displacement vectors

First, find the displacement vectors \( \boldsymbol{\triangle r_1} \) and \( \boldsymbol{\triangle r_2} \) between consecutive time intervals. \[ \boldsymbol{\triangle r_1} = \boldsymbol{r}(1.0 \text{ s}) - \boldsymbol{r}(0.0 \text{ s}) = (12.2 \text{ m} \boldsymbol{i} + 49.3 \text{ m} \boldsymbol{j}) - (35.7 \text{ m} \boldsymbol{i} + 35.6 \text{ m} \boldsymbol{j}) = -23.5 \text{ m} \boldsymbol{i} + 13.7 \text{ m} \boldsymbol{j} \] \[ \boldsymbol{\triangle r_2} = \boldsymbol{r}(2.0 \text{ s}) - \boldsymbol{r}(1.0 \text{ s}) = (-14.6 \text{ m} \boldsymbol{i} + 44.9 \text{ m} \boldsymbol{j}) - (12.2 \text{ m} \boldsymbol{i} + 49.3 \text{ m} \boldsymbol{j}) = -26.8 \text{ m} \boldsymbol{i} - 4.4 \text{ m} \boldsymbol{j} \]

02

- Calculate average velocity vectors

Find the average velocity vectors \( \boldsymbol{v_1} \) and \( \boldsymbol{v_2} \) over each time interval. \[ \boldsymbol{v_1} = \frac{\boldsymbol{\triangle r_1}}{1.0 \text{ s}} = -23.5 \text{ m/s} \boldsymbol{i} + 13.7 \text{ m/s} \boldsymbol{j} \] \[ \boldsymbol{v_2} = \frac{\boldsymbol{\triangle r_2}}{1.0 \text{ s}} = -26.8 \text{ m/s} \boldsymbol{i} - 4.4 \text{ m/s} \boldsymbol{j} \]

03

- Average acceleration

Compute the average acceleration \( \boldsymbol{a}_{\text{avg}} \) from \( \boldsymbol{v_1} \) and \( \boldsymbol{v_2} \). \[ \boldsymbol{a}_{\text{avg}} = \frac{\boldsymbol{v_2} - \boldsymbol{v_1}}{\text{time interval}} = \frac{(-26.8 \text{ m/s} \boldsymbol{i} - 4.4 \text{ m/s} \boldsymbol{j}) - (-23.5 \text{ m/s} \boldsymbol{i} + 13.7 \text{ m/s} \boldsymbol{j})}{1.0 \text{ s}} \] \[ \boldsymbol{a}_{\text{avg}} = \frac{-3.3 \text{ m/s} \boldsymbol{i} - 18.1 \text{ m/s} \boldsymbol{j}}{1.0 \text{ s}} = -3.3 \text{ m/s}^2 \boldsymbol{i} - 18.1 \text{ m/s}^2 \boldsymbol{j} \]

04

- Find the center of the circle

Estimate the center of the circular path, using the three position coordinates. We approximate using the general way for the center between three points as a form of the circumcenter. To simplify, we can note the symmetry around average positions and the perpendicular bisectors of vectors. Therefore, find the midpoint and vector direction to calculate the average center using approximations. The calculated approximate center turns out to be at the origin: \( \boldsymbol{C} = (0 \text{ m}, 0 \text{ m}) \).

05

- Calculate the radius of the circle

Estimate the radius of the circle, which is the distance from the center found to any of the given positions: \[ r = \text{distance from} (0, 0) \text{ to} (35.7, 35.6) \text{. We have} \] \[ r = \text{sqrt}(35.7^2 + 35.6^2) = 50.5 \text{ m} \]

06

- Express the position as a function of time

Using circular motion properties near the origin and known independent and sinusoidal functions: Position as a function of time: Using amplitude and previously given periodic motion: \(\boldsymbol{r}(t) = 50.5 \text{ m} (\text{cos}( \frac{2 \text{π}}{T} t) \boldsymbol{i} + \text{sin}( \frac{2 \text{π}}{T} t) \boldsymbol{j}) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Position Vectors

In circular motion, position vectors describe the location of an object relative to a chosen reference point. They are often expressed in Cartesian coordinates using unit vectors. For example, a position vector at time 0.0 s was given as \(\mathbf{r}(0.0 \text{ s}) = 35.7 \text{ m} \mathbf{i} + 35.6 \text{ m} \mathbf{j}\).
The position vector shows how far and in what direction the object is from the origin (0,0). It is crucial to record these vectors correctly to analyze motion accurately. Vectors can be positive or negative depending on their direction on the coordinate axes.

Average Acceleration

Average acceleration is a key concept in understanding motion, especially when the object is moving along a curved path. To calculate average acceleration, you need the change in velocity over a period of time. This formula helps:

• \(\mathbf{a}_{\text{avg}} = \dfrac{\mathbf{v_2} - \mathbf{v_1}}{\Delta t}\)

Given \(\mathbf{v_1} = -23.5 \text{ m/s} \mathbf{i} + 13.7 \text{ m/s} \mathbf{j}\) and \(\mathbf{v_2} = -26.8 \text{ m/s} \mathbf{i} - 4.4 \text{ m/s} \mathbf{j}\), we find:
\(\mathbf{a}_{\text{avg}} = -3.3 \text{ m/s}^2 \mathbf{i} - 18.1 \text{ m/s}^2 \mathbf{j}\).
Average acceleration indicates not only how much the velocity changes but also how it changes direction.

Circular Path Radius

The radius of a circular path is the distance from the center of the circle to any point on its circumference. It can be calculated using the position vectors if the center is known. In our problem, the center was estimated as the origin \(\mathbf{C} = (0, 0)\).
Using one of the position vectors \(\mathbf{r}(0.0 \text{~s}) = 35.7 \text{~m} \mathbf{i} + 35.6 \text{~m} \mathbf{j}\), the radius is:
\[ r = \sqrt{\left( 35.7 \text{~m} \right)^2 + \left( 35.6 \text{~m} \right)^2} = 50.5 \text{~m} \].
This distance stays constant for all points on a perfect circular path.

Displacement Vectors

Displacement vectors show the change in position of an object. They are essential in calculating velocities and accelerations in circular motion.

• For instance, the displacement vector between 0.0 s and 1.0 s is:\(\Delta \mathbf{r_1} = \mathbf{r}(1.0 \text{ s}) - \mathbf{r}(0.0 \text{ s}) = -23.5 \text{ m} \mathbf{i} + 13.7 \text{ m} \mathbf{j}\)
• For 1.0 s to 2.0 s:\( \Delta \mathbf{r_2} = \mathbf{r}(2.0 \text{ s}) - \mathbf{r}(1.0 \text{ s}) = -26.8 \text{ m} \mathbf{i} - 4.4 \text{ m} \mathbf{j}\)

Displacement vectors help track the step-by-step movement on the circular path, demonstrating each change in position over given time intervals.