Question

The velocity of the image when the object crosses the mean position and goes towards Q is

A. $A\omega$
B. $-A\omega$
C. $\dfrac{A\omega }{2}$
D. None of the above

Answer 1

This is a case of a convex lens. In this situation, the image will be executing the SHM. The object held close to the convex lens of the focal length f, executes SHM between P and Q, O being the mean position. Take x-axis as the main axis of the lens and A < D to answer the following questions.

Formula used:
For solving this question, we will be using the formula for lenses, i.e.,
$\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$

Complete step-by-step answer:
Before solving the given question, let us take a look at all the given parameters,
u = -3f
f = f
Since it’s a convex lens, the sign of the object distance is taken with negative sign
Now,
Applying the lens formula, we have
$\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$
Now, using the given values in the above formula

& \Rightarrow \dfrac{1}{v}-\dfrac{1}{-3f}=\dfrac{1}{f} \\\ \end{aligned}$$ $$\begin{aligned} & \Rightarrow \dfrac{1}{v}+\dfrac{1}{3f}=\dfrac{1}{f} \\\ \end{aligned}$$ $$\begin{aligned} & \Rightarrow \dfrac{1}{v}=\dfrac{1}{f}-\dfrac{1}{3f} \\\ \end{aligned}$$ $$\Rightarrow \dfrac{1}{v}=\dfrac{2}{3f}$$ So, we have $$\Rightarrow v=\dfrac{3f}{2}$$ Now, using the formula for the magnification $m=\dfrac{h'}{h}=\dfrac{v}{u}$ $\Rightarrow m=\dfrac{\dfrac{3f}{2}}{-3f}$ $\Rightarrow m=-\dfrac{1}{2}$ So, we can say that image will be inverted and have a path difference of $\pi $ SHM will be executed by the image in this case, $\Rightarrow y=\dfrac{A}{2}\sin (\omega t+\pi )$ On differentiating with respect to time $$\Rightarrow v=\dfrac{dy}{dt}=\dfrac{A\omega }{2}\cos (\omega t+\pi )$$ When the image crosses the mean position, t = 0 So, velocity at that time will be $\Rightarrow v=\dfrac{A\omega }{2}\cos \pi $ $\Rightarrow v=-\dfrac{A\omega }{2}$ Therefore, The velocity of the image when the object crosses the mean position and goes towards Q will be $-\dfrac{A\omega }{2}$ **So, the correct answer is “Option D”.** **Note:** You can see that the velocity of an object executing SHM can be given by the expression, $$\Rightarrow v=\dfrac{A\omega }{2}\cos (\omega t+\pi )$$ This formula will be useful in solving many such questions with ease.