Asmall block of mass m is pushed against a spring with spring constant k and held in place with a catch. the spring compresses an unknown distance x. when the catch is removed, the block leaves the spring and slides along a frictionless circular loop of radius r. when the block reaches the top of the loop, the force of the loop on the block (the normal force) is equal to twice the gravitational force on the mass. how far was the spring initially compressed?

The spring was initially compressed an distance: .

Explanation:

We need to apply the conservation of energy law to find the kinetic energy at the top of the loop. First we need to find the initial state as: where E1 is the mechanical energy at the state 1, Ek1 is the initial kinetic energy that in this case is Ek1=0 and Eu1 is the spring's potential energy and is Eu1=(1/2)kx^2. Then We need to find the final state as:. Remember that the surface is frictionless so , mechanical energy is conserved. Finally we can replace and getting: . On other side we need to apply Newton's Second Law and using the free body diagram (see attached) of the block at the top of the loop we get:. In this problem the force of the loop on the block is equal to twice the magnitude gravitational force on the block so N=2mg. Now solving this we can get: and replacing this we will find: and solving for x, we can find the spring compressed distance like: so:

no

sodium is solid at rt

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.