$\begin{aligned}
& g=\frac{G M}{R^2} \
& g^{\prime}=\frac{G(4 M)}{(2 R)^2}=g
\end{aligned}Aiscorrect,Riscorrect;butsinceT=2 \pi \sqrt{\frac{\ell}{g}}$ doesn't depend on mass ; R doesn't explain A .
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. Reason (R): The mass of the pendulum remains unchanged at Earth and the other planet. In the light of the above statements, choose the correct answer from the options given below :
Held on 22 Jan 2025 · Verified 6 Jul 2026.
(A) is false but (R) is true
(A) is true but (R) is false
Both (A) and (R) are true and (R) is the correct explanation of (A)
Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
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Match List-I with List-II. <table class="pyq-table"><tbody><tr><th>List-I</th><th>List-II</th></tr><tr><td>A. $\sin^2 \omega t$</td><td>I. Periodic with time period $T = \dfrac{\pi}{\omega}$ but not simple harmonic motion (SHM)</td></tr><tr><td>B. $\sin^3(2\omega t)$</td><td>II. Periodic with time period $T = \dfrac{2\pi}{\omega}$ but Not SHM</td></tr><tr><td>C. $\sin(\omega t) + \cos(\pi \omega t)$</td><td>III. Periodic with time period $T = \dfrac{\pi}{\omega}$ and SHM</td></tr><tr><td>D. $\cos\omega t + \cos 2\omega t$</td><td>IV. Non-periodic</td></tr></tbody></table> Choose the correct answer from the options given below :
A uniform disc of radius $R$ and mass $M$ is free to oscillate about the axis $A$ as shown in the figure. For small oscillations the time period is ______. ($g$ is acceleration due to gravity) 
The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2 = 50 - x^2$, where $x$ represents displacement. If the time period of motion is $\dfrac{x}{7}$ s, the value of $x$ is _____.
A spring stretches by $2$ mm when it is loaded with a mass of $200$ g. From equilibrium position the mass is further pulled down by $2$ mm and released. The frequency associated with the system and maximum energy in the spring are __________ Hz and __________ J, respectively. (Take g $= 10$ m/s$^2$)
As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is : $k=150 \mathrm{~N} / \mathrm{m}$ 
Work through every JEE Main Waves & Oscillations PYQ, year by year.