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A particle of mass m is attached to thre...

A particle of mass m is attached to three identical massless springs of spring constant k as shown in the m 135 figure. The time period of vertical oscillation of the particle is

A

`2pi sqrt(m/k)`

B

`2pi sqrt(m/(2k))`

C

`2pi sqrt(m/(3k))`

D

`pi sqrt(m/k)`

Text Solution

Verified by Experts

The correct Answer is:
B
When the particle of mass m at O is pushed by y in the direction of A, spring A will be compressed by y while springs B and C will be stretched by `y. = y cos 45^(@)`
The total restoring force on the mass m along AO, is

`F =F_(A) + F_(B) cos 45^(@) + F_(C ) cos 45^(@)`
`F = -ky -ky. cos 45^(@) -ky. cos 45^(@)`
`F =-[ky + ky. cos 45^(@) + ky. cos45^(@)] = -[ky + 2ky. cos 45^(@)]`
`=-[ky + 2ky cos^(2) 45^(@)]=-[ky + ky]`
`F =-2ky`........(i)
Comparing (i) with, `F = -k_(eq)y`, we get
`k_(eq) = 2k`
The time period of vertical oscillations of the particle is,
`T = 2pi sqrt(m/k_(eq)) = 2pi sqrt(m/(2k))`
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