Chapter 6.2, Problem 5CFU

To determine

To demonstrate: the given situation.

Answer to Problem 5CFU

See explanation

Explanation of Solution

Given information:

If the radius of circle is doubled and the angular angle remains same, then the linear velocity of a point on the circle is doubled.

Calculation:

Let the radius, angular angle, and linear velocity of first circle be $r,\theta ,$ and $\upsilon$ respectively.

Let the radius, central angle, length, and area of second circle be $r_2,{\theta }_2$ and ${\upsilon }_2$ respectively.

Assuming the time for both the cases is same. According to the question,

  $\begin{array}{l}{r}_2=2r\\ {\theta }_2=\theta \end{array}$

The linear velocity of first circle $\left(\upsilon \right)$ will be as follows:

  $\upsilon =r\frac{\theta }{t}$

The linear velocity of second circle $\left(\upsilon \right)$ will be as follows:

  $\begin{array}{c}{\upsilon }_2=r_2\frac{{\theta }_2}{t}\\ {\upsilon }_2=\left(2r\right)\frac{\theta }{t}\left[\text{Put the value of }{r}_2\text{ and }{\theta }_2\right]\\ {\upsilon }_2=\left(2\right)\left(\frac{r\theta }{t}\right)\\ {\upsilon }_2=2\upsilon \left[\because \upsilon =r\frac{\theta }{t}\right]\end{array}$

Thus, the linear velocity of a point on the circle is doubled if the radius of circle is doubled and the central angle remains same.