Question

Three particles P, Q and R are situated at point A on the circular path of radius 10 m. All three particles move along different paths and reach point B as shown in figure. Then the ratio of distance traversed by particles P and Q is:

• A. $\frac{3}{4}$
• B. $\frac{1}{3}$
• C. $\frac{3\pi}{4}$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (C)

(C) $\frac{3\pi}{4}$

Answer 2

The problem asks for the ratio of the distance traversed by particles P and Q. All particles start at point A and reach point B. The radius of the circular path is given as 10 m.

1. Distance traversed by particle P ($D_P$):

Particle P moves along the circular path from point A to point B. From the figure, point A is on the left side of the circle, and point B is at the bottom. Moving from A to B along the circumference in the direction indicated (counter-clockwise), the particle covers three-quarters of the circle. The total angle in a circle is $2\pi$ radians or $360^\circ$. Three-quarters of a circle corresponds to an angle $\theta = \frac{3}{4} \times 2\pi = \frac{3\pi}{2}$ radians. The distance traversed along a circular arc is given by $s = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians. Given radius $r = 10$ m. Therefore, $D_P = 10 \times \frac{3\pi}{2} = 15\pi$ m.

2. Distance traversed by particle Q ($D_Q$):

Particle Q moves along a straight line path from point A to the center O, and then from the center O to point B. The distance from A to O is the radius of the circle, so $AO = r = 10$ m. The distance from O to B is also the radius of the circle, so $OB = r = 10$ m. The total distance traversed by particle Q is the sum of these two straight line segments: $D_Q = AO + OB = r + r = 2r$. $D_Q = 2 \times 10 = 20$ m.

3. Ratio of distances traversed by particles P and Q:

The ratio is $\frac{D_P}{D_Q}$. Ratio = $\frac{15\pi}{20}$. Simplifying the fraction: Ratio = $\frac{3\pi}{4}$.

Comparing this result with the given options, option (C) matches our calculated ratio.

The final answer is $\boxed{\text{(C)}}$.

Explanation:

Particle P travels along the circumference covering 3/4 of the circle, so its distance is $r \times (3\pi/2)$. Particle Q travels along two radii (A to O and O to B), so its distance is $2r$. The ratio of these distances is $(r \times 3\pi/2) / (2r) = 3\pi/4$.