One end of a uniform rope of mass m and length l is attached to a shaft that is rotating at constant angular velocity of magnitude w. the other end is attached to a point-like object of mass m. find, the tension f in the rope as a function of r, the distance from the shaft. you may ignore the effect of gravitation.

Explanation:

Tension in the rope will be due to pseudo -force ( centrifugal force )

Centrifugal force = m rω² where m is mass being rotated and r being distance from the axis and ω is angular velocity. Naturally it will be dependent on r,  distance from the axis.

For terminal mass m attached at the end , the value of this force

F = m  ω² L

For other parts . we shall have to integrate the  values along the length because the value of m x r varies along the length.

Calculate the value of tension for  a small fraction of length dr at distance r along the length .

Mass of the fraction dr = m/L X dr

Tension due to the fraction

dT = m/L  dr X ω² r

Integrating both sides and taking limit from Lto r

T =

=

Total tension

= F +T

= + m  ω² L

=

b. lifting dumbbell weight at 20-degree angle from the body after a 45-degree angle

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