Rotational Motion

JEE Main level, single correct, medium

191 of 280

Question 191 of 280

When the position vector r⃗=xi^+yj^+zk^\vec{r}=x \hat{i}+y \hat{j}+z \hat{k} changes sign as −r⃗-\vec{r}, which one of the following vector will not flip under sign change ?

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Answer: (A) Angular momentum

The transformation r⃗→−r⃗\vec{r} \to -\vec{r} is known as a parity transformation (inversion through the origin).
Vectors that change sign under parity are called polar vectors, while those that do not change sign are called axial vectors (or pseudovectors).
Velocity is defined as v⃗=dr⃗dt\vec{v} = \frac{d\vec{r}}{dt}. Since r⃗→−r⃗\vec{r} \to -\vec{r}, we have v⃗→−v⃗\vec{v} \to -\vec{v}.
Acceleration is defined as a⃗=dv⃗dt\vec{a} = \frac{d\vec{v}}{dt}. Since v⃗→−v⃗\vec{v} \to -\vec{v}, we have a⃗→−a⃗\vec{a} \to -\vec{a}.
Linear momentum is defined as p⃗=mv⃗\vec{p} = m\vec{v}. Since v⃗→−v⃗\vec{v} \to -\vec{v}, we have p⃗→−p⃗\vec{p} \to -\vec{p}.
Angular momentum is defined as L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p}.
Under the transformation r⃗→−r⃗\vec{r} \to -\vec{r} and p⃗→−p⃗\vec{p} \to -\vec{p}, the cross product becomes:
L⃗′=(−r⃗)×(−p⃗)=r⃗×p⃗=L⃗\vec{L}' = (-\vec{r}) \times (-\vec{p}) = \vec{r} \times \vec{p} = \vec{L}.
Thus, angular momentum does not flip its sign under the transformation r⃗→−r⃗\vec{r} \to -\vec{r}.

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