What torque would have to be applied to a rod with a length of $6 m$ $6 m$ and a mass of $3 k g$ $3 kg$ to change its horizontal spin by a frequency $14 H z$ $14 Hz$ over $6 s$ $6 s$ ?

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Answer 1 · 2017-08-21T11:35:51

The torque for the rod rotating about the center is $= 131.9 N m$ $=131.9Nm$
The torque for the rod rotating one end is $= 527.8 N m$ $=527.8Nm$

The torque is the rate of change of angular momentum

$τ = \frac{d L}{d t} = \frac{d (I ω)}{d t} = I \frac{d ω}{d t}$ $tau=(dL)/dt=(d(Iomega))/dt=I(domega)/dt$

The moment of inertia of a rod, rotating about the center is

$I = \frac{1}{12} \cdot m L^{2}$ $I=1/12*mL^2$

$= \frac{1}{12} \cdot 3 \cdot 6^{2} = 9 k g m^{2}$ $=1/12*3*6^2= 9 kgm^2$

The rate of change of angular velocity is

$\frac{d ω}{d t} = \frac{14}{6} \cdot 2 π$ $(domega)/dt=(14)/6*2pi$

$= (\frac{14}{3} π) r a d s^{- 2}$ $=(14/3pi) rads^(-2)$

So the torque is $τ = 9 \cdot (\frac{14}{3} π) N m = 42 π N m = 131.9 N m$ $tau=9*(14/3pi) Nm=42piNm=131.9Nm$

The moment of inertia of a rod, rotating about one end is

$I = \frac{1}{3} \cdot m L^{2}$ $I=1/3*mL^2$

$= \frac{1}{3} \cdot 3 \cdot 6^{2} = 36 k g m^{2}$ $=1/3*3*6^2=36kgm^2$

So,

The torque is $τ = 36 \cdot (\frac{14}{3} π) = 168 π = 527.8 N m$ $tau=36*(14/3pi)=168pi=527.8Nm$

What torque would have to be applied to a rod with a length of 6m6 m and a mass of 3kg3 kg to change its horizontal spin by a frequency 14Hz14 Hz over 6s6 s?