A particle of mass m is attached to four spri | vedclass

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(A) Let the particle be displaced by a small distance $x$ towards one of the sides of the square. The springs will exert restoring forces. Considering

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$2 \pi \sqrt{\frac{m}{3k}}$

Vedclass · Answer

(A) Let the particle be displaced by a small distance $x$ towards one of the sides of the square. The springs will exert restoring forces. Considering the geometry, the effective spring constant $k_{eff}$ for the system can be determined by calculating the net restoring force $F_R$ acting on the particle. When the particle is displaced by $x$, the components of the forces from the four springs along the direction of displacement add up. The restoring force is given by $F_R = (k + k + 2k + 2k) \cdot x \cdot \cos^2(45^\circ) = (6k) \cdot x \cdot (1/2) = 3kx$. Wait, considering the vector sum of forces for a displacement $x$ along a side, the effective force constant is $k_{eff} = 3k$. Thus, the angular frequency is $\omega = \sqrt{\frac{k_{eff}}{m}} = \sqrt{\frac{3k}{m}}$. The period of oscillation is $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{3k}}$.

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