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Question #42060

Difficulty: 8/10 | 🟢 Free

The Question

A particle undergoes damped one-dimensional oscillatory motion along the $x$ -axis. Its acceleration $a$ is given by the differential equation $a = -\omega_0^2 x - b v$ , where $\omega_0$ is the natural angular frequency, $b$ is a positive damping constant, $x$ is the displacement, and $v$ is the velocity. The particle is released from rest at $x = x_0$ ( $x_0 > 0$ ) at $t = 0$ and first crosses the origin $x = 0$ at $t = t_1$ . The algebraic area under the $a-t$ graph from $t = 0$ to $t = t_1$ is $-S_1$ ( $S_1 > 0$ ), and the area under the $x-t$ graph for the same time interval is $S_2$ . Which of the following correctly expresses the damping constant $b$ ?

Options

A.
$\frac{\omega_0^2 S_2 - S_1}{x_0}$
B.
$\frac{\omega_0^2 S_2 + S_1}{x_0}$
C.
$\frac{S_1 - \omega_0^2 S_2}{x_0}$
D.
$\frac{\omega_0^2 S_2 - S_1}{2 x_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Area Under Acceleration-Time|SHM Definition via Differential Equation|Damped Oscillation Equation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to interpret the area under kinematic graphs within the context of a differential equation for damped harmonic motion.

The primary trap is attempting to use work-energy theorems or standard SHM formulas (like $v = A\omega$ ) which are invalid due to the non-conservative damping force.

Students must recognize that the area under the $a-t$ graph represents the exact integral of the equation of motion over time ( $\Delta v$ ), and similarly, $\int v dt$ yields the exact boundary values of displacement ( $\Delta x$ ).

Failing to integrate the differential equation directly and instead relying on pseudo-conservation laws will lead to a dead end.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Area Under Acceleration-Time
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free