Physics Q48 – JEE-MAIN 2026

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

\frac{x}{7}

s, the value of x is _______.

Enter Calculated Numerical Value:

Clue 1 of 3: Relating Equations

Relate the given velocity-displacement equation to the standard form for Simple Harmonic Motion (SHM) to find the motion's parameters.

🧠 Full Solution Path

Step 1: Determine the Angular Frequency (ω)✦ Active

The standard equation for the velocity $v$ of a particle in Simple Harmonic Motion (SHM) as a function of its displacement $x$ is given by $v^{2} = ω^{2} (A^{2} - x^{2})$ , where $A$ is the amplitude and $ω$ is the angular frequency.

The given equation is $v^{2} = 50 - x^{2}$ . We can rewrite the standard equation as $v^{2} = ω^{2} A^{2} - ω^{2} x^{2}$ .

By comparing the coefficients of the $x^{2}$ term in both the given and standard equations:

- ω^{2} x^{2} = - 1 \cdot x^{2}

This implies that $ω^{2} = 1$ , so the angular frequency is $ω = 1$ rad/s.

Step 2: Calculate the Time Period (T)○ Expand

The time period $T$ of SHM is related to the angular frequency $ω$ by the formula:

T = \frac{2 π}{ω}

Substituting the value $ω = 1$ rad/s, we find the time period:

T = \frac{2 π}{1} = 2 π s

Step 3: Solve for the Value of x○ Expand

The problem states that the time period of the motion is also given by the expression $T = \frac{x}{7}$ s. Note that the 'x' in this expression is a parameter to be found, not the displacement.

Equating the two expressions for the time period $T$ :

\frac{x}{7} = 2 π

Solving for x, we get $x = 14 π$ . In the context of numerical answer questions for this exam, it is common to use the approximation $π \approx \frac{22}{7}$ .

x = 14 \times \frac{22}{7} = 2 \times 22 = 44

Therefore, the value of x is 44.

💡 Teacher's Secret Hint

Be careful not to confuse the variable 'x' representing displacement with the parameter 'x' in the expression for the time period. They are distinct quantities in this problem.

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Physics Question 48 – JEE-MAIN 2026

Clue 1 of 3: Relating Equations

🧠 Full Solution Path