1. **Problem statement:** We analyze two quantities in mechanics: linear momentum $\mathbf{p}$ of a particle and the quantity $\mathbf{r} \times \mathbf{p}$, where $\mathbf{r}$ is the position vector relative to an origin. 2. **Part (a): Which quantity depends on the choice of origin?** - Linear momentum $\mathbf{p}$ is defined as $m\mathbf{v}$, where $m$ is mass and $\mathbf{v}$ is velocity. - Velocity $\mathbf{v}$ is independent of the choice of origin because it depends on the particle's motion, not on where we place the origin. - Therefore, $\mathbf{p}$ does **not** depend on the origin. - The quantity $\mathbf{r} \times \mathbf{p}$ depends on $\mathbf{r}$, the position vector relative to the origin. - Changing the origin changes $\mathbf{r}$, so $\mathbf{r} \times \mathbf{p}$ **does** depend on the choice of origin. - Examples of origin-dependent quantities: angular momentum $\mathbf{L} = \mathbf{r} \times \mathbf{p}$, torque $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$. 3. **Part (b): Which quantity does not depend on the choice of origin?** - Linear momentum $\mathbf{p}$ is independent of origin as explained. - Other examples of origin-independent quantities include kinetic energy $\frac{1}{2}mv^2$, mass $m$, and speed $|\mathbf{v}|$. 4. **Part (c): Discussion on origin dependence** - A quantity is **independent of origin** if it depends only on intrinsic properties of the particle (like velocity, mass) that do not change when the coordinate system shifts. - A quantity is **dependent on origin** if it involves position vectors measured from the origin, so changing the origin changes the vector and thus the quantity. - In summary, quantities involving absolute position vectors are origin-dependent, while those involving relative motion or intrinsic properties are origin-independent.