Question

Comprehensions type questions. A particle is moving in a circle of radius \(\mathrm{R}\) with constant speed. The time period of the particle is T Now after time \(\mathrm{t}=(\mathrm{T} / 6)\) Average speed of the particle is (A) \((\pi \mathrm{R} / 6 \mathrm{~T})\) (B) \([(2 \pi R) / 3 \mathrm{~T}]\) (C) \([(2 \pi R) / T]\) (D) \((\mathrm{R} / \mathrm{T})\)

Short answer

The short answer is: The average speed of the particle after time \(t = \frac{T}{6}\) is \(Option \ (A): \frac{\pi R}{3T}\).

Step-by-step solution

Calculate the portion of the circumference traveled

Since the particle completes one full circle in time T, and we are looking at time t = T/6, we need to find the fraction of the circumference that corresponds to this time. We can find this by using the proportion: \(\frac{\text{time period}}{\text{total time}} = \frac{\text{distance traveled}}{\text{total distance}}\) We know the total time is T and the time period given is T/6. Therefore: \[\frac{T/6}{T} = \frac{\text{distance traveled}}{2\pi R}\] Now, we need to solve for the distance traveled.

Calculate the average speed

To calculate the average speed, we will use the formula: \[\text{Average speed} = \frac{\text{distance traveled}}{\text{time taken}}\] We have the time taken as T/6. Now, we just need to solve for the distance traveled, using the proportion we set up earlier: \[\frac{T/6}{T} = \frac{\text{distance traveled}}{2\pi R}\] By cross multiplying and solving for the distance traveled, we get: \[\text{distance traveled} = \frac{(T/6)}{T} * 2\pi R = \frac{(2\pi R)}{6}\] Now, plugging the distance traveled and time taken into the average speed formula, we have: \[\text{Average speed} = \frac{\frac{(2\pi R)}{6}}{T/6}\] Simplifying, we get: \[\text{Average speed} = \frac{\pi R}{3T}\] Thus, the answer is \(Option \ (A): \frac{\pi R}{3T}\).

Key concepts

Average Speed in Circular Motion

When an object travels in a circle, understanding its average speed is crucial. The average speed is defined as the total distance traveled divided by the total time taken. In a circular motion, distance involves the path along the circumference.

For a particle moving at constant speed around a circle of radius \( R \), the full circumference is \( 2\pi R \). If the particle takes time \( T \) to complete a full circle, the time it takes to travel any portion of the circle can influence the average speed.

Let's say, in this case, we only consider the particle's travel over a sixth of the time \( \( T/6 \) \). We calculate the distanced traveled, which would be \( \rac{1}{6} \) of the circumference, or \( \frac{2\pi R}{6} \). This knowledge helps us find the average speed using the formula:

• \( \text{Average speed} = \frac{\text{distance traveled}}{\text{time taken}} \)

Substituting with values:

• \( \text{Average speed} = \frac{2\pi R / 6}{T/6} \)

After simplification, the average speed becomes \( \frac{\pi R}{3T} \). This formula tells us the speed over that specific segment of time.

The average speed shows how fast something is moving without considering directions of motion during that period. This becomes a foundation through which you better comprehend the entire motion's context.

Time Period in Circular Motion

The concept of the time period often highlights how much time it takes for a particle to complete one full cycle or rotation around a circular path. This full cycle (circle) is directly related to the formula for speed and distance in uniform circular motion.

For a particle moving uniformly around a circle of radius \( R \), its time period, \( T \), is when it returns to its original position on the circle. This time \( T \) connects to how the average speed can be calculated, since \( T \) directly links with the distance traveled, \( 2\pi R \), when counting one full circular trip.

In the scenario given, any portion of this time period can represent a specific segment of the circular motion. Here, when the particle travels for a fraction of the time period, \( T/6 \), it covers a sixth of the circle's distance.

The time period thus frames the fundamental relationship between motion distance and rotational speed:

• The complete time \( T \) results from \( \rac{2\pi R}{v} \), where \( v \) is the constant linear speed.

Thus, understanding the time period makes it easier to piece together and analyze changes or sections of circular travel, facilitating clear insights into periodic patterns in motion.

Distance Traveled in Circular Motion

In circular motion, the concept of distance traveled requires adapting how we normally view straight-line distance. Here, distance means the length along the circular path covered by a particle.

When considering a full circle of radius \( R \), the entire distance covered is the circle's circumference, given by \( 2\pi R \). But, if we want to know the distance traveled for only a portion of a full cycle, time fractions help understand sections of the path.

In the given example, the particle makes the journey of \( \frac{1}{6} \) of its full circle, since it's moving for a span of \( T/6 \) (one-sixth of the period \( T \)). The formula for the distance traveled over this part becomes:

• \( \text{distance traveled} = \frac{T/6}{T} \times 2\pi R = \frac{2\pi R}{6} \).

This distance calculation is fundamental for further determining average speed.

The concept ensures that the covered path during any time fraction can break down the uniform motion around a circle into measurable parts. So, recognizing these divisions in the circular path gives students a better grip on solving related problems about speed, time, and periodic motion.