Question

Derive second equation of rotational motion.

Answer 1

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Answer 2

Answer:

The Second equation of rotational motion is θ = ω't + $\frac{1}{2} \alpha t^{2}$

Explanation:

Considering an object revolving in a circular path, then

Angular velocity (ω) = dθ/dt

θ is the angular displacement and t is the time.

Then, ω.dt = dθ

Since, ω = (ω' + $\alpha t$ ) { This is the first equation of rotational motion }

Now, we replace  ω with (ω' + $\alpha t$ ) and we get,

(ω' + $\alpha t$ )dt =  dθ

Integrating both sides, we have,

$\int\limits^_ {} \,$ (ω' + $\alpha t$ )dt =  $\int\limits^_ {} \,$dθ

Integrating these,

we integrate ω' from the limits 0 to T

we integrate $\alpha t$ from the limits 0 to T

And we integrate dθ from the limits 0 to θ

After Integrating these, we get :

θ = ω't + $\frac{1}{2} \alpha t^{2}$

Thus, the second equation of rotational motion is θ = ω't + $\frac{1}{2} \alpha t^{2}$

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