Question 481: Centre of Mass and Rotational Motion

Question

A rigid body rotates about a fixed axis with variable angular velocity $(\alpha-\beta t)$ at time $t$, where $\alpha$ and $\beta$ are constants. The angle through which it rotates before it comes to rest is

Accepted Answer

Angular velocity of rigid body
$$ \omega(t)=\alpha-\beta t $$
when body is at rest
$$ \begin{aligned} & \omega=\alpha-\beta t=0 \ \Rightarrow & t=\frac{\alpha}{\beta} \end{aligned} $$
$$ \begin{aligned} &	ext { Angle rotated is the integral of angualr velocity. }\ &\begin{aligned} & 	heta=\int_0^{\alpha \beta} \alpha(t) d t=\int_0^{\alpha \beta}(\alpha-\beta t) d t \ & 	heta=\left[\alpha t-\frac{\beta t^2}{2}ight]_0^{\alpha / \beta} \ & 	heta=\frac{\alpha^2}{\beta}-\frac{\alpha^2}{2 \beta} \ & 	heta=\frac{\alpha^2}{2 \beta} \end{aligned} \end{aligned} $$

Answer

$\frac{\alpha}{\beta}$

Answer

$\frac{\alpha^2}{\beta}$

Answer

$\frac{\alpha^2}{2 \beta}$

Answer

$\frac{\alpha}{2 \beta}$

Centre of Mass and Rotational Motion

A rigid body rotates about a fixed axis with variable angular velocity (\alpha-\beta t) at time t, where \alpha and \beta are constants. The angle through which it rotates before it comes to rest is

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