Obtain $\tau = I\alpha $ from angular momentum of rigid body.
Obtain $\tau = I\alpha $ from angular momentum of rigid body.
The total angular momentum of a motion of rigid body about a fixed axis $\overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{L}_{z}}+\overrightarrow{\mathrm{L}_{\perp}}$
Differentiating w.r.t. time
$\frac{d \overrightarrow{\mathrm{L}}}{d t}=\frac{d \overrightarrow{\mathrm{L}}_{z}}{d t}+\frac{d \overrightarrow{\mathrm{L}_{\perp}}}{d t}$
but $\frac{d \overrightarrow{\mathrm{L}_{z}}}{d t}=\tau \hat{k}$ and $\frac{d \overrightarrow{\mathrm{L}_{\perp}}}{d t}=0$
$\therefore \frac{d \overrightarrow{\mathrm{L}}}{d t}=\tau \hat{k}$
but $\overrightarrow{\mathrm{L}_{\mathrm{Z}}}=\mathrm{I} \omega \hat{k}$
$\therefore \frac{d(\mathrm{I} \omega)}{d t}=\tau \hat{k}$
$\mathrm{I} \frac{d \omega}{d t}=\tau$
$\therefore \quad \overrightarrow{\mathrm{I} \alpha}=\vec{\tau} \Rightarrow \vec{\tau}=\mathrm{I} \vec{\alpha}$
Scalar form $\tau=\mathrm{I} \alpha$ is similar to $\mathrm{F}=m a$ in linear motion.
Similar Questions
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In the List-$I$ below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \neq \beta$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{ p }$ is the linear momentum, $\bar{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-$I$ with those quantities in List-$II$, which are conserved for that path.
List-$I$
List-$II$
$P$ $\dot{r}(t)=\alpha t \hat{t}+\beta t \hat{j}$
$1$ $\overrightarrow{ p }$
$Q$ $\dot{r}(t)=\alpha \cos \omega t \hat{i}+\beta \sin \omega t \hat{j}$
$2$ $\overrightarrow{ L }$
$R$ $\dot{r}(t)=\alpha(\cos \omega t \hat{i}+\sin \omega t \hat{j})$
$3$ $K$
$S$ $\dot{r}(t)=\alpha t \hat{i}+\frac{\beta}{2} t^2 \hat{j}$
$4$ $U$
$5$ $E$
In the List-$I$ below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \neq \beta$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{ p }$ is the linear momentum, $\bar{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-$I$ with those quantities in List-$II$, which are conserved for that path.
| List-$I$ | List-$II$ |
| $P$ $\dot{r}(t)=\alpha t \hat{t}+\beta t \hat{j}$ | $1$ $\overrightarrow{ p }$ |
| $Q$ $\dot{r}(t)=\alpha \cos \omega t \hat{i}+\beta \sin \omega t \hat{j}$ | $2$ $\overrightarrow{ L }$ |
| $R$ $\dot{r}(t)=\alpha(\cos \omega t \hat{i}+\sin \omega t \hat{j})$ | $3$ $K$ |
| $S$ $\dot{r}(t)=\alpha t \hat{i}+\frac{\beta}{2} t^2 \hat{j}$ | $4$ $U$ |
| $5$ $E$ |
- [IIT 2018]
A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion is the $x-y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by $\vec{L}_O$ and $\vec{L}_P$ respectively, then
A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion is the $x-y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by $\vec{L}_O$ and $\vec{L}_P$ respectively, then
- [IIT 2012]