Question

When the position vector \[ \vec r = x\hat{i}+y\hat{j}+z\hat{k} \] changes sign as \(\vec r \rightarrow -\vec r\), which one of the following vectors will {not flip under sign change?}

• A. Linear momentum
• B. Angular momentum
• C. Velocity
• D. Acceleration

Hint:

Vector cross products of two vectors that both change sign remain unchanged.

Answer 1

The Correct Option is B

The problem provides a position vector \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) and describes a sign change where \(\vec{r} \rightarrow -\vec{r}\). This indicates flipping the direction of the vector.

We need to determine which of the provided vectors do not flip under this sign change. Vectors that are linear with direct dependence on position, such as velocity or linear momentum, will flip because changing the sign of the position vector implies changing the direction.

• Linear Momentum: Defined as \(\vec{p} = m\vec{v}\), where \(m\) is mass and \(\vec{v}\) is velocity. Since velocity changes to \(-\vec{v}\) when position changes to \(-\vec{r}\), linear momentum also flips direction.
• Velocity: Directly derived from the position vector (change in position over time), so when \(\vec{r} \rightarrow -\vec{r}\), the velocity also reverses.
• Acceleration: Defined as the change in velocity over time. Acceleration remains the same under the flipping of \(\vec{r}\) as it is a second derivative in nature, with respect to time. However, in this scenario, it was expected to compare primary dependencies to position vector changes—linear versus angular indication here helps.
• Angular Momentum: Defined as \(\vec{L} = \vec{r} \times \vec{p}\). Under \(\vec{r} \to -\vec{r}\) and \(\vec{p} \to -\vec{p}\), the cross product \(\vec{L} = (-\vec{r}) \times (-\vec{p}) = \vec{r} \times \vec{p}\) does not flip, given that a double negative in cross product retains the direction.

Thus, the vector that does not flip under the sign change \(\vec{r} \rightarrow -\vec{r}\) is Angular Momentum.