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Chapter 4: Problem 16

An athlete completes half a round of a circular track of radius \(\mathrm{R}\), then the displacement and distance covered by the athlete are a. \(2 R\) and \(\pi R\) b. \(\pi R\) and \(2 R\) c. \(R\) and \(2 \pi R\) d. \(2 \pi R\) and \(R\)

Short Answer

Expert verified
The answer is (a): Displacement is \(2R\), distance is \(\pi R\).

Step by step solution

01

Understanding Displacement

First, understand that displacement is the shortest straight line distance between the starting and ending point of the athlete. If the track is circular, half a round would make the athlete travel from one point on the circumference to the directly opposite point on the circle. Thus, the displacement here is a straight line passing through the center of the circle, which is equal to the diameter of the circle. Since the diameter is twice the radius, the displacement is equal to \(2R\).
02

Calculating the Distance Covered

The distance covered by the athlete is the length of the path traveled. As the athlete completes half a circle, this path length is half the circumference of the circle. The circumference of a circle is given by \(2\pi R\), so half of it is \(\pi R\). Thus, the distance covered by the athlete is \(\pi R\).
03

Matching the Answer Choices

Now, we compare our calculated displacement \(2R\) and distance \(\pi R\) with the given options. The correct match is option (a), where the distance covered is \(\pi R\) and the displacement is \(2R\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Displacement in Circular Motion
In circular motion, displacement refers to the shortest path connecting two points on the circle. Imagine an athlete running on a circular track, completing exactly half a round. The runner begins at one point on the circular boundary and finishes at the point directly opposite, across the center of the circle.
  • The displacement is essentially the diameter of the circle, which is the straight line passing through the center from one side to the other.
  • The formula for the diameter is simply twice the radius (\(2R\)).
This concept is a visual and straightforward representation, showcasing how displacement is about direction and linear distance, not the path taken.
Distance in Circular Motion
Distance in circular motion describes the actual path length traveled by a moving object around a circle. Unlike displacement, distance accounts for every bend and curve along the journey. For an athlete running halfway around a circular track, this is the arced path on the boundary of the circle.
  • The circumference of a complete circle is given by the formula \(2\pi R\).
  • Therefore, completing half a circle equates to covering half the circumference, resulting in a distance of \(\pi R\).
This measurement ensures that even indirect paths account for every meter traveled by the athlete.
Circumference of a Circle
The term "circumference" refers to the total boundary length of a circle. Understanding circumference is crucial in problems involving circular motion, like determining distances covered on circular tracks.
  • Circumference is calculated using the formula \(C = 2\pi R\), where \(R\) is the radius of the circle.
  • This formula arises from the definition of \(\pi\), as the ratio of the circle's circumference to its diameter.
Comprehending how to calculate circumference is essential for analyzing scenarios when motion around a circle is involved, helping us understand half or partial circumferences as well.
Geometry in Physics
Geometry plays a vital role in physics, especially when analyzing motion along various paths. Circular motion combines geometric principles with dynamic scenarios, showing how shapes influence physical situations.
  • In circular motion, geometry helps us define and calculate distances and displacements based on circle properties, like radius or diameter.
  • It also aids in visualizing and solving problems where objects move along curved paths, ensuring we can accurately represent and quantify motion.
By integrating geometry into physics, we gain valuable insights into the structure and behavior of moving objects, bridging the gap between theoretical shapes and real-world motion.

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Most popular questions from this chapter

A particle moves with uniform velocity. Which of the following statements about the motion of the particle is true? a. Its speed is zero b. Its acceleration is zero c. Its acceleration is opposite to the velocity d. Its speed may be variable.

The distances moved by a freely falling body (starting from rest) during \(1^{\text {st }}, 2^{\text {od }}, 3^{\text {rd }} \ldots ., n^{\text {th }}\) second of its motion are proportional to a. even numbers b. odd numbers c. all integral numbers d. squares of integral numbers

If displacement of a particle is zero, the distance covered a. must be zero b. may or may not be zero c. cannot be zero d. depends upon the particle

Between two stations a train starting from rest first accelerates uniformly, then moves with constant velocity and finally retards uniformly to come to rest. If the ratio of the time taken be \(1: 8: 1\) and the maximum speed attained be \(60 \mathrm{~km} / \mathrm{h}\), then what is the average speed over the whole journey? a. \(48 \mathrm{~km} / \mathrm{h}\) b. \(52 \mathrm{~km} / \mathrm{h}\) c. \(54 \mathrm{~km} / \mathrm{h}\) d. \(56 \mathrm{~km} / \mathrm{h}\)

Between the two stations, a train accelerates from rest uniformly at first, then moves with constant velocity and finally retards uniformly to come to rest. If the ratio of the time taken be \(1: 8: 1\) and the maximum speed attained be \(60 \mathrm{~km} / \mathrm{h}\), then what is the average speed over the whole journey? a. \(48 \mathrm{~km} / \mathrm{h}\) b. \(52 \mathrm{~km} / \mathrm{h}\) c. \(54 \mathrm{~km} / \mathrm{h}\) d. \(56 \mathrm{~km} / \mathrm{h}\)
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