Question

Comprehensions type questions. A particle is moving in a circle of radius $R$ with constant speed. The time period of the particle is T Now after time $\mathrm{t}=(\mathrm{T}/6)$ Average velocity of the particle is (A) $(3\mathrm{R}/\mathrm{T})$ (B) $(6\mathrm{R}/\mathrm{T})$ (C) $(2\mathrm{R}/\mathrm{T})$ (D) $(4\mathrm{R}/\mathrm{T})$

Short answer

The average velocity of the particle after time t = T/6 is (B) $\frac{6R}{T}$.

Step-by-step solution

In order to find the displacement, we need to find the angle covered by the particle after time t = T/6. Since the time period of the particle is T, it means that it covers 360 degrees or $2\pi$ radians in time T. Therefore, to find the angle covered after time t, we can use the formula for simple harmonic motion: $$\theta =\frac{2\pi }{T}\times t$$ #Step 2: Calculate the angle covered after time t = T/6#

Now, we will plug in the given value for time t = T/6 in the above formula to find the angle covered at this specific time: $$\theta =\frac{2\pi }{T}\times \frac{T}{6}$$ $$\theta =\frac{2\pi }{6}$$ $$\theta =\frac{\pi }{3}$$ #Step 3: Calculate the displacement#

Since the particle is moving in a circle of radius R, we need to find the position of the particle on the circumference after time t. The position of the particle can be calculated using the sine and cosine of the angle covered. The displacement of the particle can then be calculated using the Pythagorean theorem as follows: $$d=2R\sin \frac{\theta }{2}$$ #Step 4: Calculate the displacement after an angle of θ = π/3 radians#

Now, we'll plug θ = π/3 into the displacement formula: $$d=2R\sin \frac{\pi /3}{2}$$ $$d=2R\sin \frac{\pi }{6}$$ Since $\sin (\pi /6)=1/2$, we get $$d=R$$ #Step 5: Calculate the average velocity#

Average velocity can be calculated by dividing the displacement by the time taken, which in this case is t = T/6. We will now find the average velocity using the calculated displacement: $$v_{avg}=\frac{R}{T/6}$$ #Step 6: Simplify the average velocity expression#

We will now simplify the expression obtained in the last step: $$v_{avg}=\frac{R}{T/6}$$ $$v_{avg}=\frac{R\times 6}{T}$$ $$v_{avg}=\frac{6R}{T}$$ Thus, the average velocity of the particle after time t = T/6 is (B) $\frac{6R}{T}$.

Key concepts

Circular Motion

Circular motion is a type of movement where an object travels along the circumference of a circle. Here, the speed of the particle remains constant throughout the journey. Imagine it like a merry-go-round or a spinning wheel, where the path is the circle, and every part of the circle is identical in terms of movement.

In circular motion, two important quantities to consider are:

• Radius (R): This is the distance from the center of the circle to any point along its circumference.
• Speed: Though speed remains constant, the direction changes continuously, differentiating it from linear motion.

Understanding these concepts helps in determining other properties like displacement and average velocity when an object follows this type of path.

Displacement

Displacement is not just about how far an object travels but where it ends up in comparison to where it started. In the original problem, we calculated displacement after a portion of the circular path was covered.

Key points about displacement include:

• Direction Specific: Displacement considers only the shortest path between the initial and final positions, which can sometimes be tricky in circular paths.
• Pythagorean Theorem: Often in circular motion, using the sine and cosine of the angle can help determine this short distance effectively.
• Example: For our example with an angle of $\pi /3$, the displacement was found to be $R$, the radius of the circle.

Recognizing the difference between distance and displacement is key, especially in circular motions where the two are usually not the same.

Simple Harmonic Motion

Simple harmonic motion represents oscillatory movement in a straight path, often described by a sinusoidal pattern. It is critical to understand because the angle calculation in circular movements often uses formulas from simple harmonic motion.

Important aspects about simple harmonic motion are:

• Periodic Motion: This means the motion repeats itself over a regular interval, similar to how a particle travels around the circle repeatedly.
• Equations of Motion: The formula $\theta =\frac{2\pi }{T}\times t$ helps us determine the angle covered, borrowing from simple harmonic motion principles.
• Application: Recognizing angles and translating that to physical movements, such as displacement, mimics the back-and-forth nature of simple harmonic motion.

Relating circular motion to simple harmonic motion equations make it easier to analyze rotational systems like our original exercise.

Time Period

The time period is the time an object takes to complete one full revolution around the circle. It's important as it tells us how fast a particle moves within its circular path.

This concept involves:

• Full Rotation: For a particle in circular motion, it signifies the time to travel 360 degrees or $2\pi$ radians around the circle.
• Time Segments: In our exercise, the time segment $t=T/6$ helps us calculate how far the particle moves in a fractional part of a full time period.
• Speed and Period: With constant speed, knowing the time period allows for calculating other factors like average velocity over specific intervals.

Grasping the time period helps in determining the dynamics of the motion, enabling you to understand concepts like velocity and displacement in such systems.