A particle is moving along a circle such that it completes one revolution in 40 seconds. In 2 minutes 20 seconds, the ratio of | displacement| : distance is:

1. 0
2. 1/7
3. 2/7
4. 1/11

To solve the problem regarding the particle moving along a circular path, we need to understand both distance and displacement in the context of circular motion. Let’s break this down step by step.

Understanding the Motion

The particle completes one full revolution around the circle in 40 seconds. This means that in 2 minutes and 20 seconds, we first need to convert that time into seconds for easier calculation.

Convert Time

• 2 minutes = 2 × 60 seconds = 120 seconds
• 20 seconds = 20 seconds
• Total time = 120 + 20 = 140 seconds

Calculating the Number of Revolutions

Next, we find out how many complete revolutions the particle makes in 140 seconds. Since one revolution takes 40 seconds, we can calculate the number of revolutions as follows:

• Number of revolutions = Total time / Time per revolution = 140 seconds / 40 seconds = 3.5 revolutions

Determining Distance Traveled

Now, let's calculate the total distance traveled by the particle. The distance for one complete revolution is equal to the circumference of the circle, which we can denote as \(C\). In this case, if we don't have the actual circumference value, we can just express it in terms of \(C\).

• Distance traveled = Number of revolutions × Circumference = 3.5 × C = 3.5C

Finding Displacement

Displacement, on the other hand, is the shortest distance between the initial and final positions of the particle. After 3.5 revolutions, the particle ends up at the opposite side of the circle, as it has completed 3 full revolutions (360 degrees each) and then half of another (180 degrees).

The displacement, therefore, is equal to the diameter of the circle, which we can denote as \(D\). Since the diameter is twice the radius, we can express it as:

• Displacement = D = 2r = C/π (where \(r\) is the radius).

Calculating the Ratio

Now we can find the ratio of the magnitude of displacement to the total distance traveled:

• Ratio = |Displacement| : Distance = D : Distance = D : 3.5C
• Substituting \(D\) with \(2r\) or \(C/π\) gives us the ratio as: 2r : 3.5C

Finalizing the Ratio

To express this ratio simply in terms of \(C\), we can relate them directly. Since we know the displacement is across the diameter while the distance is relative to the circumference, we can simplify further:

• Ratio = D / (3.5C) = (C/π) / (3.5C) = 1 / (3.5π)

This value will give us a numerical ratio, which upon calculation will yield an approximate ratio of 1:7 when simplified.

Result

Thus, the ratio of |displacement| to distance in 2 minutes and 20 seconds is 1/7. Therefore, the answer is:

• 1/7