JEE Main SHM PYQ

6.(JEE Main 2023 (Online) 24th January Morning Shift )

JEE Main 2023 (Online) 24th January Morning Shift Physics - Simple Harmonic Motion Question 12 English

A block of a mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is $\frac{π}{\sqrt{x}}$ in SI unit. The value of $x$ is ____________.

Correct Answer is (5)

Both the springs are in parallel so net spring constant is $K_{net} = K_{1} + K_{2} = 40 N / m$

So $T = 2 π \sqrt{\frac{m}{K_{n e t}}}$

$\begin{aligned} = 2 π \sqrt{\frac{2}{40}} \\ = \frac{π}{\sqrt{5}} \end{aligned}$

$∴ x = 5$

7. (JEE Main 2022 (Online) 25th July Morning Shift)

JEE Main 2022 (Online) 25th July Morning Shift Physics - Simple Harmonic Motion Question 31 English

In figure $(A)$ , mass ' $2 m^{'}$ is fixed on mass ' $m$ ' which is attached to two springs of spring constant $k$ . In figure (B), mass ' $m$ ' is attached to two springs of spring constant ' $k$ ' and ' $2 k^{'}$ . If mass ' $m$ ' in (A) and in (B) are displaced by distance ' $x^{'}$ horizontally and then released, then time period $T_{1}$ and $T_{2}$ corresponding to $(A)$ and (B) respectively follow the relation.

A. $\frac{T_{1}}{T_{2}} = \frac{3}{\sqrt{2}}$

B. $\frac{T_{1}}{T_{2}} = \sqrt{\frac{3}{2}}$

C. $\frac{T_{1}}{T_{2}} = \sqrt{\frac{2}{3}}$

D. $\frac{T_{1}}{T_{2}} = \frac{\sqrt{2}}{3}$

Correct Answer is Option (A)

Both the springs are in parallel combination in both the diagrams so

$T_{1} = 2 π \sqrt{\frac{3 m}{2 k}}$

and $T_{2} = 2 π \sqrt{\frac{m}{3 k}}$

So, $\frac{T_{1}}{T_{2}} = \frac{3}{\sqrt{2}}$

8. (JEE Main 2022 (Online) 24th June Evening Shift)

Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :

A. 1 : 2

B. 3 : 2

C. 3 : 1

D. 2 : 3

Correct Answer is Option (B)

$ω_{1} A_{1} = ω_{2} A_{2}$

$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{ω_{2}}{ω_{1}}$

$= \sqrt{\frac{k_{2}}{m_{2}}} \times \sqrt{\frac{m_{1}}{k_{1}}} = \sqrt{\frac{9 k}{100} \times \frac{50}{2 k}} = \frac{3}{2}$

9.(JEE Main 2022 (Online) 27th July Morning Shift )

A mass $0.9 kg$ , attached to a horizontal spring, executes SHM with an amplitude $A_{1}$ . When this mass passes through its mean position, then a smaller mass of $124 g$ is placed over it and both masses move together with amplitude $A_{2}$ . If the ratio $\frac{A_{1}}{A_{2}}$ is $\frac{α}{α - 1}$ , then the value of $α$ will be ___________.

Correct Answer is (16)

$(0.9) A_{1} \sqrt{\frac{K}{0.9}} = (0.9 + 0.124) A_{2} \sqrt{\frac{K}{0.9 + 0.124}}$

$\frac{A_{1}}{A_{2}} = \sqrt{\frac{0.9 + 0.124}{0.9}}$

$= \sqrt{\frac{1.024}{0.9}}$

$= \frac{α}{α - 1}$

$α = 16$

10.(JEE Main 2022 (Online) 26th July Evening Shift )

As per given figures, two springs of spring constants $k$ and $2 k$ are connected to mass $m$ . If the period of oscillation in figure (a) is $3 s$ , then the period of oscillation in figure (b) will be $\sqrt{x} s$ . The value of $x$ is ___________.

JEE Main 2022 (Online) 26th July Evening Shift Physics - Simple Harmonic Motion Question 29 English

Correct Answer is (2)

For case (a),

$K_{e q} = \frac{2 K}{3}$

For case (b),

$K_{e q} = 3 K$

$∵$ $T = 2 π \sqrt{\frac{m}{K}}$

$∴$ $\frac{T_{a}}{T_{b}} = \sqrt{\frac{K_{b}}{K_{a}}}$

$\frac{3}{T_{b}} = \sqrt{\frac{3 K \times 3}{2 K}} = \frac{3}{\sqrt{2}}$

$T_{b} = \sqrt{2}$

$x = 2$