A 242-g block is pressed against a spring of force constant 1.62 kN/m until the block compresses the spring 10.0 cm. The spring rests at the bottom of a ramp inclined at 60.0° to the horizontal. Using energy considerations, determine how far up the incline the block moves from its initial position before it stops under the following conditions. (a) if the ramp exerts no friction force on the block (b) if the coefficient of kinetic friction is 0.440

A) d = 3.94m

B) d' = 3.14m

Explanation:

We are given;

Mass of block; m = 242g = 0.242 kg

Spring constant; k = 1.62 kN/m = 1.62 x 10³ N/m

Extension of spring; x = 10 cm =0.1m

Coefficient of friction;µ = 0.44

Angle of inclination; θ = 60°

A) Equating the energy at the bottom of the ramp to the energy at a distance d up the ramp (at a height h in the vertical direction) we find;

W = (1/2)kx² = (1/2)•1.62 x 10³•0.1² = 8.1 J

W = mgh = 8.1

Where, h = d•sinθ

Thus, 8.1 = 0.242 x 9.8 x d sin 60

8.1 = 2.05387d

d = 8.1/2.05387

d = 3.94m

B) Here we equate the initial energy, still 8.1J, at the bottom of the ramp to the total energy at a distance d' up the ramp (at a height h') which is the sum of the gravitational potential energy (at h') and the energy losses due to

friction. The energy losses due to friction are calculated from the work

done by friction which is the force of friction times the displacement of

the object. The force of friction is simply the product of the normal force and the coefficient of kinetic friction. The normal force is found through a free body diagram and has magnitude F_n = mg cos θ . Putting this all together, we have;

8.1 = mgh' + ΔE

Thus,

8.1 = mgd'•sin θ + µmgd'•cos θ

So, d' = 8.1/[mgd'•sin θ + µmgd'•cos θ]

Plugging in relevant values;

8.1/[0.242x9.8(d'sin60 + (0.44 x d' x cos60))

8.1 = 2.3716d'(0.866 + 0.22)

8.1 = d'(2.3716 x 1.086)

d' = 8.1/2.576 = 3.14m

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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.