A wooden block of mass 0.5 kg and density 800 kg m^ -3 is attached...

Since the block is completely submerged in water, the volume of water displaced remains constant. Therefore, the buoyant force acting on the block is constant throughout its motion.

The constant buoyant force and the weight of the block only determine the equilibrium position of the spring. The restoring force for the simple harmonic motion is provided solely by the spring.

The effective spring constant of the system is $k_{eff} = k = 50 \text{ N m}^{-1}$.

The time period of vertical oscillations of a spring-mass system is given by:

$T = 2\pi \sqrt{\dfrac{m}{k}}$

Substituting the given values, $m = 0.5 \text{ kg}$ and $k = 50 \text{ N m}^{-1}$:

$T = 2\pi \sqrt{\dfrac{0.5}{50}}$

$T = 2\pi \sqrt{\dfrac{1}{100}}$

$T = 2\pi \times \dfrac{1}{10}$

$T = \dfrac{\pi}{5} \text{ s}$