Understanding Displacement in Circular Motion

This problem asks us to find the displacement of a boy completing rounds on a circular track. It's important to understand what displacement means in physics, especially compared to distance.

• Displacement: This is a vector quantity representing the shortest straight-line distance between the initial and final positions of an object. It depends only on the starting and ending points, not the path taken.
• Distance: This is a scalar quantity representing the total length of the path traveled by an object.

In circular motion, if an object completes a full round and returns to its starting point, the displacement is zero, even though the distance covered is the circumference of the circle.

Analyzing the Given Information

We are given the following details about the boy's motion on the circular track:

• Diameter of the circular track: \(D = 200 \text{ m}\)
• Time taken for one complete round: \(T = 30 \text{ s}\)
• Total time of motion: \(t_{\text{total}} = 3 \text{ minutes and } 45 \text{ seconds}\)

We need to find the displacement at the end of the total time.

Step-by-Step Calculation of Displacement

Step 1: Convert Total Time to Seconds

The time per round is given in seconds, so let's convert the total time into seconds:

\(3 \text{ minutes} = 3 \times 60 \text{ s} = 180 \text{ s}\)

Total time in seconds \(t_{\text{total}} = 180 \text{ s} + 45 \text{ s} = 225 \text{ s}\)

Step 2: Calculate the Number of Rounds Completed

To find the number of rounds completed, we divide the total time by the time taken for one round:

\(\text{Number of rounds} = \frac{\text{Total time}}{\text{Time per round}}\)

\(\text{Number of rounds} = \frac{225 \text{ s}}{30 \text{ s}}\)

\(\text{Number of rounds} = 7.5\)

This means the boy completes 7 full rounds and then an additional 0.5 (half) of a round.

Step 3: Determine the Final Position

After completing any whole number of rounds (like 7 full rounds), the boy returns to his original starting position. For these full rounds, the displacement is zero.

The displacement at the end of 225 seconds is determined by the position after the fractional part of the motion, which is 0.5 rounds.

0.5 rounds means the boy covers half the circumference of the circle. If the boy starts at point A on the circle, completing half a round will take him to the point diametrically opposite to A.

Step 4: Calculate the Displacement

The displacement is the straight-line distance between the initial position (point A) and the final position (the point diametrically opposite to A). This straight-line distance is equal to the diameter of the circular track.

Displacement = Diameter of the track

Displacement = \(200 \text{ m}\)

Therefore, at the end of 3 minutes and 45 seconds, the boy's displacement is 200 m.

Revision Table: Key Values and Results

Additional Information on Displacement and Circular Motion

Understanding displacement is crucial in kinematics. Here are a few more points:

• Displacement is a vector quantity, meaning it has both magnitude and direction. In this problem, the magnitude is 200 m, and the direction would be from the starting point towards the ending point across the diameter.
• Distance traveled in 7.5 rounds would be \(7.5 \times (\text{Circumference})\). The circumference of the track is \(\pi D = \pi \times 200 \text{ m}\). So, distance \( = 7.5 \times 200\pi \text{ m} = 1500\pi \text{ m}\). This is significantly different from the displacement.
• For any motion that ends at the starting point, the displacement is always zero, regardless of how far the object traveled.
• In circular motion, if an object completes a half round, its displacement is equal to the diameter. If it completes a quarter round, the displacement is the hypotenuse of a right triangle with sides equal to the radius, which is \(r\sqrt{2}\).