Animated Solution for Physics - Rotational Motion: The time dependence of the position of a particle of mass $m=2$ is given by $\mathbf{r}(t)=2t\hat{\mathbf{i}}-3t^2\hat{\mathbf{j}}$. Its angular momentum, with respect to the origin, at time $t=2$ is

Imagine a particle tracing a curved path in the $xy$-plane. We are given its position vector as a function of time: $\mathbf{r}(t)=2t\hat{\mathbf{i}}-3t^2\hat{\mathbf{j}}$. Our mission is to find its angular momentum about the origin at a specific moment, $t=2$ seconds.

Angular momentum, denoted by $\mathbf{L}$, is a measure of the amount of rotation an object has, taking into account its mass, shape, and speed. For a point particle, it is elegantly defined by the cross product of its position vector $\mathbf{r}$ and its linear momentum $\mathbf{p}$.

Mathematically, this is written as:

To use our master equation, we first need the velocity vector $\mathbf{v}$. Velocity is simply the rate at which position changes with time. So, we differentiate the position vector $\mathbf{r}(t)$ with respect to $t$:

Applying basic differentiation rules, we get:

Now, we substitute our expressions for $\mathbf{r}$ and $\mathbf{v}$ into the angular momentum formula:

Let's expand this cross product carefully. Remember the rules for unit vectors: the cross product of a vector with itself is zero ($\hat{\mathbf{i}}\times \hat{\mathbf{i}}=0$, $\hat{\mathbf{j}}\times \hat{\mathbf{j}}=0$). Also, $\hat{\mathbf{i}}\times \hat{\mathbf{j}}=\hat{\mathbf{k}}$ and $\hat{\mathbf{j}}\times \hat{\mathbf{i}}=-\hat{\mathbf{k}}$.

We have a beautiful, simplified expression for the angular momentum at any time $t$. The problem asks for the value at $t=2$ seconds for a particle of mass $m=2$ kg. Let's plug these numbers in:

Pro Tip: You could also substitute $t=2$ into $\mathbf{r}$ and $\mathbf{v}$ before taking the cross product. At $t=2$, $\mathbf{r}=4\hat{\mathbf{i}}-12\hat{\mathbf{j}}$ and $\mathbf{v}=2\hat{\mathbf{i}}-12\hat{\mathbf{j}}$. Then, $\mathbf{L}=2[(4\hat{\mathbf{i}}-12\hat{\mathbf{j}})\times (2\hat{\mathbf{i}}-12\hat{\mathbf{j}})]=2[-48\hat{\mathbf{k}}-(-24\hat{\mathbf{k}})]=2[-24\hat{\mathbf{k}}]=-48\hat{\mathbf{k}}$. Both paths lead to the same elegant result!