A particle is executing SHM along a straight line. Its velocities at distances and from the mean position are and , respectively. Its time period is [2015]

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Q.
A particle is executing SHM along a straight line. Its velocities at distances $x_{1}$ and $x_{2}$ from the mean position are $V_{1}$ and $V_{2}$ , respectively. Its time period is [2015]

1 $2 π \sqrt{\frac{V_{1}^{2} + V_{2}^{2}}{x_{1}^{2} + x_{2}^{2}}}$

2 $2 π \sqrt{\frac{V_{1}^{2} - V_{2}^{2}}{x_{1}^{2} - x_{2}^{2}}}$

3 $2 π \sqrt{\frac{x_{1}^{2} + x_{2}^{2}}{V_{1}^{2} + V_{2}^{2}}}$

4 $2 π \sqrt{\frac{x_{2}^{2} - x_{1}^{2}}{V_{1}^{2} - V_{2}^{2}}}$

Ans.
(4)

In SHM, velocities of a particle at distances $x_{1}$ and $x_{2}$ from mean position are given by

$V_{1}^{2} = ω^{2} (a^{2} - x_{1}^{2})$ ...(i)

$V_{2}^{2} = ω^{2} (a^{2} - x_{2}^{2})$ ...(ii)

From equations (i) and (ii), we get

$V_{1}^{2} - V_{2}^{2} = ω^{2} (x_{2}^{2} - x_{1}^{2})$

$ω = \sqrt{\frac{V_{1}^{2} - V_{2}^{2}}{x_{2}^{2} - x_{1}^{2}}}$ $\Rightarrow T = 2 π \sqrt{\frac{x_{2}^{2} - x_{1}^{2}}{V_{1}^{2} - V_{2}^{2}}}$

Solution

Q.
A particle is executing SHM along a straight line. Its velocities at distances $x_{1}$ and $x_{2}$ from the mean position are $V_{1}$ and $V_{2}$ , respectively. Its time period is [2015]

1 $2 π \sqrt{\frac{V_{1}^{2} + V_{2}^{2}}{x_{1}^{2} + x_{2}^{2}}}$

2 $2 π \sqrt{\frac{V_{1}^{2} - V_{2}^{2}}{x_{1}^{2} - x_{2}^{2}}}$

3 $2 π \sqrt{\frac{x_{1}^{2} + x_{2}^{2}}{V_{1}^{2} + V_{2}^{2}}}$

4 $2 π \sqrt{\frac{x_{2}^{2} - x_{1}^{2}}{V_{1}^{2} - V_{2}^{2}}}$

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Solution

Q. A particle is executing SHM along a straight line. Its velocities at distances x1 and x2 from the mean position are V1 and V2, respectively. Its time period is [2015]

1 2πV12+V22x12+x22

2 2πV12-V22x12-x22

3 2πx12+x22V12+V22

4 2πx22-x12V12-V22

Ans. (4) In SHM, velocities of a particle at distances x1 and x2 from mean position are given by V12=ω2(a2-x12) ...(i) V22=ω2(a2-x22) ...(ii) From equations (i) and (ii), we get V12-V22=ω2(x22-x12) ω=V12-V22x22-x12⇒ T=2πx22-x12V12-V22

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Q.
A particle is executing SHM along a straight line. Its velocities at distances $x_{1}$ and $x_{2}$ from the mean position are $V_{1}$ and $V_{2}$ , respectively. Its time period is [2015]

1 $2 π \sqrt{\frac{V_{1}^{2} + V_{2}^{2}}{x_{1}^{2} + x_{2}^{2}}}$

2 $2 π \sqrt{\frac{V_{1}^{2} - V_{2}^{2}}{x_{1}^{2} - x_{2}^{2}}}$

3 $2 π \sqrt{\frac{x_{1}^{2} + x_{2}^{2}}{V_{1}^{2} + V_{2}^{2}}}$

4 $2 π \sqrt{\frac{x_{2}^{2} - x_{1}^{2}}{V_{1}^{2} - V_{2}^{2}}}$