Question

A particle moves horizontally in a uniform circular motion, over a horizontal xyplane. At one instant, it moves through the point at coordinates$\left(4.00m,4.00m\right)$ with a velocity of$\mathbf{-}\mathbf{5}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$ and an acceleration of$\mathbf{+}\mathbf{12}\mathbf{.}\mathbf{5}\hat{\mathbf{j}}\mathbf{}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ (a)What are the x coordinates of the center of the circular path? (b)What are the y coordinates of the center of the circular path?

Short answer

1. x coordinate of the center of the circular path x = 4.00 m
2. y coordinate of the center of the circular path y = 6.00 m

Step-by-step solution

Given data

1. The coordinates of point P( 4.00 m, 4.00 m)
2. The velocity is, $\overrightarrow{v}=-5.00\hat{i}m/s$
3. The acceleration is $\overrightarrow{a}=+12.5\hat{j}m/s^2$

Understanding the concept of circular motion

Using the equation for acceleration, we can find the radius of the circular path. The particle is moving with uniform circular motion. Hence, the magnitude of velocity and acceleration is constant. The acceleration of the particle is positive and going towards the center. When we draw the figure, the coordinates of the center of the circular path will be (x,y+r).

Formula:

$a=\frac{v^2}{r}$

(a) Calculate the x coordinates of the center of the circular path

The particle is moving in a circular path with uniform circular motion, hence the centripetal acceleration is

$$\begin{gathered} a=\frac{v^2}{r} \\ r=\frac{v^2}{a} \\ =\frac{{(-5.00m/s)}^2}{12.5m/s^2} \\ =2.00m \end{gathered}$$

From the figure, let P( 4.00 m, 4.00 m ) be any point on the circular path, C be the center of the circle, and its coordinates will be C ( x, y + r ). From the figure, x coordinate is the same as P point.

Therefore, the x coordinate of the center of the circular path is 4.00 m .

(b) Calculate the y coordinates of the center of the circular path

The acceleration is positive, so it is intheupward direction at that point. Then the y coordinate of the center will be the y coordinate of P plus the circle’s radius.

$$\begin{gathered} \mathrm{C}(\mathrm{x},\mathrm{y}+\mathrm{r})=\mathrm{C}(4.00\mathrm{m},6.00\mathrm{m}) \\ \mathrm{y}=6.00\mathrm{m} \end{gathered}$$

Therefore, the y coordinate of the center of the circular path is 6.00 m .