A ball of mass m is attached to a string of length L. It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion. Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical circle, the ball's speeds are vt andvb, and the corresponding tensions in the string are Tt and Tb. Tt and Tb have magnitudes Tt and find Tb-Tt, the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle. Express the difference in tension in terms of m and g.

Explanation:

When the ball is at the bottom position of the vertical circle, the forces acting on the ball are:

- The tension in the string, , upward

- The weight of the ball, , downward

The resultant of these forces must be equal to the centripetal force, which points upward as well (towards the center of the circle), so:

(1)

where is the speed of the ball at the bottom of the circle, r the radius of the circle, and m the mass of the ball.

When the ball is at the top position of the vertical circle, the weight still acts downward, however the tension in the string also acts downward, so the equation of the forces becomes:

(2)

where

is the tension in the string in the top position

is the speed of the ball in the top position

By subtracting eq.(2) from eq.(1), we find:

So, this is the difference in tension between the two positions.

what are the options.

explanation:

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.