Question

A particle is moving in a circle of radius \(\mathrm{R}\) with constant speed. It covers an angle \(\theta\) in some time interval. Find displacement in this interval of time. (A) \(2 \mathrm{R} \cos (\theta / 2)\) (B) \(2 \operatorname{Rsin}(\theta / 2)\) (C) \(2 \mathrm{R} \cos \theta\) (D) \(2 \mathrm{R} \sin \theta\)

Short answer

The displacement of the particle during the time interval is \(\Delta AB = 2R\sin(\theta/2)\). Therefore, the correct answer is (B) \(2R \sin(\theta/2)\).

Step-by-step solution

Sketch the situation

Draw a circle with radius R, along with the initial position, final position, and the angle theta that the particle covers during the time interval. Label the initial position as A, the final position as B, and the center of the circle as O.

Find the displacement

In order to find the displacement (∆AB), we can use the law of cosines for triangles, which states that for a triangle with sides a, b, c and opposite angles A, B, C, we have: \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\) In this problem, the triangle is OAB, where OA and OB are the radii of the circle, so OA = OB = R, and the angle AOB is given by theta. We want to find the length of the side AB, so we can apply the law of cosines: \(\Delta AB^2 = R^2 + R^2 - 2R \cdot R \cdot \cos(\theta)\)

Simplify the equation and find the value of displacement

Simplify the equation, we get: \(\Delta AB^2 = 2R^2 - 2R^2 \cdot \cos(\theta)\) Take the square root of both sides: \(\Delta AB = \sqrt{2R^2 - 2R^2 \cdot \cos(\theta)}\) Factor out the 2R^2: \(\Delta AB = R\sqrt{2(1 - \cos(\theta))}\) Using the trigonometric identity \(\cos(\theta) = 1 - 2\sin^2(\theta/2)\), we can rewrite the displacement as: \(\Delta AB = R\sqrt{2 \cdot 2\sin^2(\theta/2)}\) Further simplifying: \(\Delta AB = 2R\sin(\theta/2)\) So the displacement of the particle during the time interval is: \(\Delta AB = 2R\sin(\theta/2)\) Therefore, the correct answer is (B) \(2R \sin(\theta/2)\).

Key concepts

Displacement in Circular Motion

When we talk about circular motion, it generally involves an object moving along the circumference of a circle. The term "displacement" in this context refers to the shortest path between the initial and final positions of the object moving in the circle. Despite the object following a curved path, the displacement is a straight line segment between these two points.
For the given problem where a particle covers an angle \( \theta \) in time, this path forms an isosceles triangle with the circle's center, where the two equal sides are the radii \( R \). The line segment forming the base of this triangle between the positions A and B on the circle is the displacement. Understanding that in circular motion, displacement is not the arc but rather the straight line connecting the start and end point is key.
Thus, in the scenario given, the displacement is calculated by effectively utilizing these geometric and trigonometric considerations.

Law of Cosines

The Law of Cosines is an important tool in geometry that helps to determine an unknown side of a triangle when two sides and the included angle are known. For a triangle with sides \( a, b, \) and \( c \) opposite the angles \( A, B, \) and \( C \) respectively, the law is given by:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
In solving our problem, we apply the Law of Cosines to the triangle formed in the circular motion scenario (triangle OAB). With both sides \( OA \) and \( OB \) as radii \( R \), and the included angle \( \theta \) at the circle's center, the Law of Cosines helps us find the shortest distance between points A and B. This distance, side AB, represents our displacement in motion.
This method, which leverages both the known values and the angle \( \theta \), simplifies the task of finding the displacement, emphasizing the power of the Law of Cosines in physics problem-solving.

Trigonometric Identities

Trigonometric identities are crucial in simplifying the expressions we deal with in both physics and mathematics. One vital identity is useful in our circular motion problem: \( \cos(\theta) = 1 - 2\sin^2(\theta/2) \). This identity allows us to express \( \cos(\theta) \) in terms of \( \sin \), which can often make calculations simpler and easier.
In practice, this particular identity transforms the expression within the square root into a multiple of \( \sin^2(\theta/2) \). This transformation is crucial when simplifying our equation further to find the displacement. Such substitutions using trigonometric identities not only help in manual calculations but also provide deeper insights into the relationships between different trigonometric functions.
Understanding and effectively utilizing these identities can greatly aid in both problem-solving and analysis for various physics challenges.

Physics Problem Solving

Physics problem solving often combines mathematical reasoning with conceptual understanding. The case of motion in a circle showcases how various tools, such as sketches, geometry, and trigonometry, come together to solve a problem.
To tackle any physics problem effectively, the following strategies are often useful:

• Break the problem into manageable parts by sketching diagrams to visualize the situation.
• Identify and apply relevant formulas and laws, like the Law of Cosines for our triangle.
• Use known principles or identities to simplify and solve equations.
• Double-check calculations by understanding the physical implications and ensure they make sense.

With practice, these strategies help students become more proficient problem solvers, capable of approaching even complex scenarios with confidence and a systematic method of reasoning.