A block of mass m is connected to two indentical springs of spring constant k which are in turn connected to fixed supports as shown in the figure. Find the time period for small oscillations of the block.

Options

A. `2pi sqrt(m/k)`
B. `2pisqrt(m/(2k))`
C. `2pisqrt((2m)/k)`
D. `pisqrt(m/(2k))`

Answer

B

Explanation

(b) Let the mass m be displaced by a small distance x to the right from its mean position as shown in figure.
(b) Due to it the spring on the left side gets starched by a length x while that on the right side gets compressed by the same length. The forces acting on the mass are `F_(1) = kx` towards left hand are `F_(2) = kx` towards left hand are
The net forces acting on the mass is, `F =F_(1) + F_(2) =- 2kx`
Here, `F prop x` and `-ve` sign shows that force is toward the mean position, therefore the motion executed by the particle is simple harmonic. Its acceleration is: `F = F_(1) + F_(2) = - 2kx`
The standard equation of SHM is:
`a = - omega^(2)x`
Comparing (i) and (ii), we get:
`omega^(2) = (2k)/m` or `omega = sqrt((2k)/m)`
Time period, `T = (2pi)/omega = 2pi sqrt(m/(2k))`