MHT CET · Physics · Rotational Motion
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 \(]\)
- A \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{r}} \times \vec{\omega}, \vec{\alpha}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}, \vec{\tau}=\) \(\overrightarrow{\mathrm{f}} \times \overrightarrow{\mathrm{r}}\)
- B \(\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}, \vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{r}}, \vec{\tau}=\) \(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}\)
- C \(\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}, \vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}, \vec{\tau}=\) \(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}\)
- D \(\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}, \vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{r}}, \vec{\tau}=\) \(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}\)
Answer & Solution
Correct Answer
(C) \(\overrightarrow{\mathrm{v}}=\vec{\omega} \times \overrightarrow{\mathrm{r}}, \vec{\alpha}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}}, \vec{\tau}=\) \(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{f}}\)
Step-by-step Solution
Detailed explanation
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.
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.
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