A particle executes simple harmonic motion between x = - A and x = + A .The time taken by it to go from O to A /2 is T $_{1}$ and to go from A /2 to A is T $_{2}$ . Then

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Answer 1

(a) The equation of SHM, when it starts swinging from the mean position, is $y = A s i n ω t$ .
At mean position, y=0, t=0 .....(1)

$\frac{A}{2} = A s i n ω t_{2}$ or $s i n ω t_{2} = \frac{1}{2} = s i n \frac{π}{6}$

$ω t_{2} = \frac{π}{6}$ or $t_{2} = \frac{π}{6 ω}$ .....(2)

At y=A time for travel from 0 to A= $t_{3}$

A=Asin $ω t_{3}$

or sin $ω t_{3} = 1 = s i n \frac{π}{2}$ or $t_{3} = \frac{π}{2 ω}$ .....(3)

$T_{1} = t_{2} - t_{1} = \frac{π}{6 ω}$ .....(4)

$T_{2} = t_{3} - t_{2} = \frac{π}{2 ω} - \frac{π}{6 ω} = \frac{π}{3 ω}$ .....(5)

$\frac{T _{1}}{T _{2}} = \frac{\frac{π}{6 ω}}{\frac{π}{3 ω}} = \frac{3 ω}{6 ω} = \frac{1}{2}$ or $\frac{T _{1}}{T _{2}} = \frac{1}{2}$

$T_{1} < T_{2}$

Answer 2

(a) The equation of SHM, when it starts swinging from the mean position, is

y = A s i n ω t

.

At mean position, y=0, t=0 .....(1)

\frac{A}{2} = A s i n ω t_{2}

or

s i n ω t_{2} = \frac{1}{2} = s i n \frac{π}{6}

ω t_{2} = \frac{π}{6}

or

t_{2} = \frac{π}{6 ω}

.....(2)

At y=A time for travel from 0 to A=

t_{3}

A=Asin

ω t_{3}

or sin

ω t_{3} = 1 = s i n \frac{π}{2}

or

t_{3} = \frac{π}{2 ω}

.....(3)

T_{1} = t_{2} - t_{1} = \frac{π}{6 ω}

.....(4)

T_{2} = t_{3} - t_{2} = \frac{π}{2 ω} - \frac{π}{6 ω} = \frac{π}{3 ω}

.....(5)

\frac{T _{1}}{T _{2}} = \frac{\frac{π}{6 ω}}{\frac{π}{3 ω}} = \frac{3 ω}{6 ω} = \frac{1}{2}

or

\frac{T _{1}}{T _{2}} = \frac{1}{2}

T_{1} < T_{2}

A particle executes simple harmonic motion between x = - A and x = + A .The time taken by it to go from O to A /2 is T $_{1}$ and to go from A /2 to A is T $_{2}$ . Then

T $_{1}$ < T $_{2}$

T $_{1}$ > T $_{2}$

T $_{1}$ = T $_{2}$

T $_{1}$ = 2 T $_{2}$

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