In case of rotational dynamics, which one of the following statements is correct? $[\vec{\omega}=$ angular…

In case of rotational dynamics, which one of the following statements is correct? $[\vec{\omega}=$ angular velocity, $\vec{v}=$ linear velocity $\overrightarrow{\mathrm{r}}=$ radius vector, $\vec{\alpha}=$ angular acceleration $\overrightarrow{\mathrm{a}}=$ linear acceleration, $\overrightarrow{\mathrm{L}}=$ angular momentum $\overrightarrow{\mathrm{p}}=$ linear momentum, $\vec{\tau}=$ torque, $\overrightarrow{\mathrm{f}}=$ centripetal force $]$
  1. $\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{r}} \times \vec{\omega}, \vec{\alpha}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}, \vec{\tau}=\overrightarrow{\mathrm{f}} \times \overrightarrow{\mathrm{r}}$
  2. $\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}, \vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{r}}, \vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}$
  3. $\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}, \vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}, \vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}$
  4. $\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}, \vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{r}}, \vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}$

Solution

Linear velocity \(\overrightarrow{(\mathrm{v})}\) : The linear velocity of a point in rotational motion is given by the cross product of the angular velocity vector \((\vec{\omega})\) and the radius vector \(\overrightarrow{(r)}\) : \(\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}\) This shows the perpendicular relationship between linear velocity and both the radius vector and the angular velocity. Angular acceleration (\(\vec{\alpha}\)): The angular acceleration and its relation to linear acceleration can often be represented with: \(\vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}\) Though, if specified to mean tangential component in a circular path, it directly relates to the tangential acceleration. Angular momentum \((\overrightarrow{\mathrm{L}})\) : The angular momentum of a particle with respect to a point is defined as: \(\overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}\) where \(\overrightarrow{\mathrm{p}}\) is the linear momentum (\(m v\)) of the particle. Torque \((\vec{\tau})\) : Torque is defined as the cross product of the radius vector \(\overrightarrow{(r)}\) and the force vector \(\overrightarrow{(\mathbf{f})}\) : \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}\) This defines the rotational effect of a force applied at a distance from a pivot. Each component is consistent with the right-hand rule and the classical definitions in rotational dynamics.

Asked in: MHT CET 2024 (03 May Shift 1)

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