a weight with a mass `m` is attached to the end of a string with a length of length `l` fastened to a vertical rod rtating with a angular velocity `ω`.another string of the same length as the first and carrying on its end another weight with a mass `m` secured to the first weight.
prove that during rotation,the angle between the first string and the vertical will be smaller than the angle between the vertical and the second string.disregard the weight of the string.
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2 Answers
1we can get answer directly from fbd and writing the equations
let tension in thread 1 be T1 and in Thread 2 be T2
and let angles with the horizontal be θ1 and θ2
then as they both are at a constant height and a constant distance from the pole
we can write the equations
T1cosθ1=mw2r1+T2cosθ2
and T1sinθ1=T2sinθ2=mg
and T2cosθ2=mw2r2
we see that T1cosθ1>T2cosθ2
but T1sinθ1=T2sinθ2
squaring the above equation
T12(1-cos2θ1)=T22(1-cos2θ2)
so we get
T12-T22=(T1cosθ1)2-(T2cosθ2)2
as T1cosθ1>T2cosθ2
T1>T2
but T1sinθ1=T2sinθ2
as T1>T2
sinθ1<sinθ2
so θ1<θ2
