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Chapter 2: Problem 325

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

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

Step by step solution

01

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#
02

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#
03

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) \(\frac{6R}{T}\).

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Most popular questions from this chapter

If \(\mathrm{f}\) is the frequency of a body moving in a circular path with constant speed. a is its centrifugal acceleration, so. (A) \(\mathrm{a} \propto \mathrm{f}\) (B) a \(\propto \mathrm{f}^{2}\) (C) \(\mathrm{a} \propto \mathrm{f}^{3}\) (D) a \(\propto(1 / \mathrm{f})\)

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