A spring with spring constant 40 N/m is attached to the ceiling, and a 5.1-cm-diameter, 1.5 kg metal cylinder is attached to its lower end. The cylinder is held so that the spring is neither stretched nor compressed, then a tank of water is placed underneath with the surface of the water just touching the bottom of the cylinder. When released, the cylinder will oscillate a few times but, damped by the water, quickly reach an equilibrium position.
When in equilibrium, what length of the cylinder is submerged?

24.5 cm

Explanation:

Given

Force constant of spring, k = 40 N/m

Diameter of spring, d = 5.1 cm = 0.051 m

Mass of cylinder, m = 1.5 kg

Let us assume that the cylinder is hanging in such a way that the circular end is parallel with the water. Also, we assume that the tank water level is not materially affected by the displacement of the cylinder while the cylinder sinks. The water is fresh and as we all know, the density of water is 1000 kg/m³

To solve this, we assume x to be the spring extension and it's equivalent sinking distance(in meters). We then apply the formula,

mg = kx + ρgAx

mg = x(k + ρgA)

x = mg / (k + ρgA), where

A = πd²/4

A = (3.142 * 0.051²)/4

A = 0.0082 / 4

A = 0.00205 m²

x = 1.5 * 9.81 / [40 + (1000 * 9.81 * 0.00205)]

x = 14.715 / (40 + 20.1105)

x = 14.715 / 60.1105

x = 0.245 m

or 24.5 cm of stretch or sinking

scattering of light by tiny particles in air to produce the colors we see on sky, is dependent on the following factors:

(i) wavelength of light

(ii) size of the scattering particle  

(iii) distance traveled by light through the atmosphere.

according to rayleigh's law of scattering, the amount of scattering is inversely proportional to the fourth power of its wavelength. shorter the wavelength, greater is its scattering. hence violet/blue light is scattered more than red.

sun's light is a composite light ( visible light) with wavelengths in the range 400 nm (violet) to 700 nm (red). as sunlight passes through our atmosphere, the molecules of air scatter light according to their wavelengths. during sunrise and sunset, the sun's position is below the horizon. hence light from the sun travels a greater distance in the atmosphere when compared to its position a noon.

as sunlight enters the earth's atmosphere, blue light is scattered away and at the end of its journey in the sky, only red and orange wavelengths remain in the beam.

hence at sunrise and sunset, the sky appears reddish orange.

dang that balls went far