Step 1: The angular momentum \( \mathbf{L} \) of a particle about the origin is given by the cross product:
\[ \mathbf{L} = \mathbf{r} \times m \mathbf{u} \]
where \( \mathbf{r} \) is the position vector and \( \mathbf{u} \) is the velocity vector.
Step 2: The magnitude of the angular momentum is given by:
\[ L = |\mathbf{r}| \cdot m |\mathbf{u}| \cdot \sin \theta \]
where \( \theta \) is the angle between \( \mathbf{r} \) and \( \mathbf{u} \).
Step 3: Since \( b \) is constant and the particle moves in a straight line, the angular momentum varies with \( \theta \), and the correct expression is:
\[ L = |\mathbf{r}| \cdot |\mathbf{u}| \cdot \sin \theta. \]
A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{\mathrm{x}}{5}}$ where $\mathrm{x}=$ _______.
If $\overrightarrow{\mathrm{L}}$ and $\overrightarrow{\mathrm{P}}$ represent the angular momentum and linear momentum respectively of a particle of mass ' $m$ ' having position vector $\overrightarrow{\mathrm{r}}=\mathrm{a}(\hat{\mathrm{i}} \cos \omega \mathrm{t}+\hat{\mathrm{j}} \sin \omega \mathrm{t})$. The direction of force is
Which of the following are correct expression for torque acting on a body?
A. $\ddot{\tau}=\ddot{\mathrm{r}} \times \ddot{\mathrm{L}}$
B. $\ddot{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\ddot{\mathrm{r}} \times \ddot{\mathrm{p}})$
C. $\ddot{\tau}=\ddot{\mathrm{r}} \times \frac{\mathrm{d} \dot{\mathrm{p}}}{\mathrm{dt}}$
D. $\ddot{\tau}=\mathrm{I} \dot{\alpha}$
E. $\ddot{\tau}=\ddot{\mathrm{r}} \times \ddot{\mathrm{F}}$
( $\ddot{r}=$ position vector; $\dot{\mathrm{p}}=$ linear momentum; $\ddot{\mathrm{L}}=$ angular momentum; $\ddot{\alpha}=$ angular acceleration; $\mathrm{I}=$ moment of inertia; $\ddot{\mathrm{F}}=$ force; $\mathrm{t}=$ time $)$
Choose the correct answer from the options given below: