Question

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

• A. $2\pi \sqrt{\frac{m}{k}}$
• B. $2\pi \sqrt{\frac{m}{2k}}$
• C. $2\pi \sqrt{\frac{m}{3k}}$
• D. $\pi \sqrt{\frac{m}{k}}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{m}{2k}}$

When the particle of mass m at O is pushed by y in the direction of A, the spring A will be compressed by y while spring B and C will be stretched by $y^{\prime }=ycos{45}^{\circ }$. So, that the total restoring force on the mass m is along OA

$F_{net}=F_A+F_{B}cos{45}^{\circ }+F_{C}cos{45}^{\circ }$

$=ky+2k{y}^{\prime }cos{45}^{\circ }$

$=ky+2k\left(ycos{45}^{\circ }\right)cos{45}^{\circ }$

$=2ky$

Also, $F_{net}=k^{\prime }y\Rightarrow k^{\prime }y=2ky\Rightarrow k^{\prime }=2k$

$T=2\pi \sqrt{\frac{m}{k^{\prime }}}=2\pi \sqrt{\frac{m}{2k}}$