A particle is making simple harmonic motion along the X-axis. If at a distances x₁ and x₂ from the mean position the velocities of the particle are $υ_{1} a n d υ_{2}$ respectively. The time period of its oscillation is given as:

Option 1 -

T = $2 π \sqrt[]{\frac{x_{2}^{2} - x_{1}^{2}}{υ_{1}^{2} - υ_{2}^{2}}}$

Option 2 -

$T = 2 π \sqrt[]{\frac{x_{2}^{2} + x_{1}^{2}}{υ_{1}^{2} - υ_{2}^{2}}}$

Option 3 -

$T = 2 π \sqrt[]{\frac{x_{2}^{2} - x_{1}^{2}}{υ_{1}^{2} + υ_{2}^{2}}}$

Option 4 -

$T = 2 π \sqrt[]{\frac{x_{2}^{2} + x_{1}^{2}}{υ_{1}^{2} + υ_{2}^{2}}}$

2 Views | Posted a month ago

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1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

a month ago

Correct Option - 1

Detailed Solution:

As we know that $υ^{2} = ω^{2} (A^{2} - x^{2})$ $^{}^{}^{}^{}$ for SHM, so

$υ_{1}^{2} = ω^{2} (A^{2} - x_{1}^{2}) . . . . . . . (i), a n d υ_{2}^{2} = ω^{2} (A^{2} - x_{2}^{2}) . . . . . . . . . (i i)$

Subtracting equation (ii) from equation (i), we have

$υ_{1}^{2} - υ_{2}^{2} = ω^{2} (x_{2}^{2} - x_{1}^{2}) \Rightarrow ω = \frac{2 π}{T} = \sqrt[]{\frac{υ_{1}^{2} - υ_{2}^{2}}{x_{2}^{2} - x_{1}^{2}}} \Rightarrow T = 2 π \sqrt[]{\frac{x_{2}^{2} - x_{1}^{2}}{υ_{1}^{2} - υ_{2}^{2}}}$

As we know that <math> <mrow> <msup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </math> <math><mrow><msup><mrow></mrow><mrow></mrow></msup><msup><mrow></mrow><mrow></mrow></msup><mrow><mrow><msup><mrow></mrow><mrow></mrow></msup><msup><mrow></mrow><mrow></mrow></msup></mrow></mrow></mrow></math> for SHM, so <math> <mrow> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>A</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> Subtracting equation (ii) from equation (i), we have <math> <mrow> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>⇒</mo> <mi>ω</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mi>T</mi> </mrow> </mfrac> <mo>=</mo> <mroot> <mrow> <mfrac> <mrow> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> <mo>⇒</mo> <mi>T</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mroot> <mrow> <mfrac> <mrow> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mrow> <mi>υ</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> </mrow> </math>

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A particle is making simple harmonic motion along the X-axis. If at a distances x₁ and x₂ from the mean position the velocities of the particle are $υ_{1} a n d υ_{2}$ respectively. The time period of its oscillation is given as:

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A particle is making simple harmonic motion along the X-axis. If at a distances x1 and x2 from the mean position the velocities of the particle are υ 1 a n d υ 2 respectively. The time period of its oscillation is given as:

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A particle is making simple harmonic motion along the X-axis. If at a distances x₁ and x₂ from the mean position the velocities of the particle are $υ_{1} a n d υ_{2}$ respectively. The time period of its oscillation is given as: