Question

A particle of mass m is attatched to three springs A, B and C of equal force constants k as shown in figure (12−E6). If the particle is pushed slightly against the spring C and released, find the time period of oscillation.
Figure

Answer 1

Explanation:

• We know that time period of oscillation, $\mathrm{T}=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}}$.  If the particle is pushed slightly against the spring C and released, the spring A and B tries to pull back the particle.  
• Thus,  the total force on the particle is $F=F 1+F 2$where F1 is force due to spring C and F2 is the force due to spring A and B.  
• Thus, the resultant force $\mathrm{F}=2 \mathrm{k} \mathrm{x}$. We know that acceleration $a=\frac{F}{m}=\frac{2 k x}{m}.$ From the above formula, the time period is $=\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{m}}{2 \mathrm{k}}}$.