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Simple Harmonic Motion question

2023 · 13 Apr · Shift 1 · Q45
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  5. /2023 · 13 Apr · Shift 1 · Q45

Simple Harmonic Motion question

2023 · 13 Apr · Shift 1 · Q45

JEE MainPhysicsSimple Harmonic MotionMCQ+4 / −1
Which graph represents the difference between total energy and potential energy of a particle executing SHM vs it's distance from mean position ?
  1. A
    JEE Main 2023 (Online) 13th April Morning Shift Physics - Simple Harmonic Motion Question 34 English Option 1
  2. B
    JEE Main 2023 (Online) 13th April Morning Shift Physics - Simple Harmonic Motion Question 34 English Option 2
  3. C
    JEE Main 2023 (Online) 13th April Morning Shift Physics - Simple Harmonic Motion Question 34 English Option 3
  4. D
    JEE Main 2023 (Online) 13th April Morning Shift Physics - Simple Harmonic Motion Question 34 English Option 4
View written solutionFree

Correct answer: A

  1. For a particle executing SHM, the total mechanical energy is constant:

E=12kA2E = \frac{1}{2}kA^2E=21​kA2

where AAA is the amplitude.

  1. The potential energy at displacement xxx from the mean position is:

U=12kx2U = \frac{1}{2}kx^2U=21​kx2

  1. Therefore, the difference between total energy and potential energy is:

E−U=12kA2−12kx2E-U = \frac{1}{2}kA^2 - \frac{1}{2}kx^2E−U=21​kA2−21​kx2

E−U=12k(A2−x2)E-U = \frac{1}{2}k(A^2-x^2)E−U=21​k(A2−x2)

  1. But in SHM,

K=E−UK = E-UK=E−U

so this quantity is just the kinetic energy.

  1. Hence,

E−U∝A2−x2E-U \propto A^2-x^2E−U∝A2−x2

This is a downward-opening parabola in xxx.

  1. Key features of the graph:
  • Maximum at x=0x=0x=0
  • Zero at x=±Ax=\pm Ax=±A
  • Symmetric about x=0x=0x=0
  • Opens downward

So the correct graph must be an inverted parabola between −A-A−A and +A+A+A.

Therefore, the correct option is A.

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