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WBJEEA thin uniform rod, pivoted at \(O\), is rotating in the horizontal plane...
A thin uniform rod, pivoted at \(O\), is rotating in the horizontal plane with constant angular speed \(\omega\), as shown in the figure. At time \(t=0\), a small insect starts from \(O\) and moves with constant speed \(v\), with respect to the rod towards the other end. It reaches the end of the rod at \(t=T\) and stops. The angular speed of the system remains \(\omega\) throughout. The magnitude of the torque \((|\vec{\tau}|)\) about \(O\), as a function of time is best represented by which plot?
Solution:
Angular momentum, \(|\mathbf{L}|\) or \(L=I \omega\) (about axis of rod) Moment of inertia of the rod-insect system. \(I=I_{\text {rod }}+m x^{2}=I_{\text {rod }}+m v^{2} t^{2}\) Here, \(m=\) mass of insect \(\therefore \quad L=\left(I_{\mathrm{rod}}+m v^{2} t^{2}\right) \omega\)
Now \(|\tau|=\frac{d L}{d t}=\left(2 m v^{2} t \omega\right)\) or \(|\tau| \propto t\)
i.e. the graph is straight line passing through origin.
After time \(T, L=\) constant
\(\therefore \quad|\tau| \text { or } \frac{d L}{d t}=0\)
i.e., when the insect stops moving, \(L\) does not change and therefore \(T\) becomes constant.
Now \(|\tau|=\frac{d L}{d t}=\left(2 m v^{2} t \omega\right)\) or \(|\tau| \propto t\)
i.e. the graph is straight line passing through origin.
After time \(T, L=\) constant
\(\therefore \quad|\tau| \text { or } \frac{d L}{d t}=0\)
i.e., when the insect stops moving, \(L\) does not change and therefore \(T\) becomes constant.
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