Question
Obtain an expression for the torque acting on a rotating body with constant angular acceleration. Hence state the dimensions and SI unit of torque.
✓
Answer
For $m_1, a_1=r_1 \alpha$
For $m_2, a_2=r_2 \alpha$
For $m_n, a_n=r_n \alpha$
$ f _1= m _1 a _1= m _1 r _1 \alpha$
$f _2= m _2 a _2= m _2 r _2 \alpha$
$f _{ n }= m _{ n } r _{ n } \alpha $
$\text { Torque }(\vec{\tau})=\vec{r} \times \vec{f}$
$=\operatorname{rfsin} 90^{\circ}$
$\tau= rf$
$\tau_1= rf _1= m _1 r _1^2 \alpha$
$\tau_2= m _2 r _2^2 \alpha$
$\tau_{ n }= m _{ n } r _{ n }^2 \alpha$
$\tau=\tau_1+\tau_2+\ldots+\tau_n$ $\text { Total Torgue on the body, } \vec{\tau}_{\text {net }}=\vec{\tau}_1+\vec{\tau}_2+\vec{\tau}_3+\ldots \vec{\tau}_{ n }$
$= m _1 r _1^2 \alpha+ m _2 r _2^2 \alpha+\ldots .+ m _{ n } r _{ n }^2 \alpha$
$=\alpha\left( m _1 r _1^2+ m _2 r _2^2+\ldots .+ m _{ n } r _{ n }^2\right)$
$I = mr ^2$
$\vec{\tau}_{\text {net }}=\left( I _1+ I _2+ I _3+\ldots . . I _n\right) \alpha$
$= I \alpha$
$\text { Unit: N.m }$
$\text { dimension: }\left[ ML ^2 T ^{-2}\right]$
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