Morin problem 942 Pivot and string orin problem A stick of m

Morin, problem 9.42: Pivot and string. orin. problem A stick of mass m and length l spins with frequency w around an axis. The stick makes an angle with the axis. One end is pivoted on the axis, and the other end is connected to the axis by a string that is perpendicular to the axis. What is the tension in the string, and what is the force that the pivot applies to the stick? (Ignore gravity.)

Solution

balancing moment about pivot point,

due to rotation rod will feel centrifugal force.

let take a small dl length part at distance l .
distance from axis (or radius), r = l sin@

mass of this part = (M/L)dl

so force on this part, df = (M/L)dl w^2 l sin@

moment due to this force = ((M/L) dl w^2 l sin@ )l cos@

integrating from 0 to L.

torque1 = (M w^2 sin@ cos@ L^3) / (3L) = Mw^2 sin@ cos@ L^2 / 3

torque due to Tension about pivot point = LTcos@

balancing net moment,

Mw^2 sin@ cos@ L^2 / 3 = L T cos@

T = M w^2 L sin@ / 3 ................Ans

force on small part, df = (M/L)dl w^2 l sin@

integrating it from 0 to L to get net centripetal force on rod.

F = M w^2 L^2 sin@ / 2L = M w^2 L sin@ / 2

and now balancing force in horizontal direction on rod.

T + Fv - F = 0

Fv = F - T = Mw^2 L sin@ [ 1/2 - 1/3 ]

Fv = M w^2 L sin@ / 6 .........Ans force pivot applied.