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Step-by-Step Explanation
Step 1: Identify the physical situation
Consider a bob of mass $m$ attached to a light, inextensible string of length $l$, suspended from a fixed support. The bob moves in a horizontal circular path with a constant angular speed $\omega$ about the vertical.
Step 2: Understand what is being asked
We need to analyze how the angular momentum of the bob about its point of suspension changes. Specifically, we want to know whether its magnitude or direction (or both) change over time.
Step 3: Express angular momentum about the suspension point
The linear velocity $\vec{v}$ of the bob is tangential to the horizontal circle, and the position vector $\vec{r}$ of the bob (from the suspension point) has length $l$. Since the motion is with uniform angular speed $\omega$, the speed of the bob is $v = \omega \cdot r_\perp$, where $r_\perp$ is the horizontal component of the string length that contributes to circular motion (in this case, for small angles, approximately $r_\perp \approx l$, but the exact geometry involves the horizontal radius of the circle).
The angular momentum of the bob about the point of suspension is given by
$$
\vec{L} = \vec{r} \times m \vec{v}.
$$
The magnitude of $\vec{r}$ remains constant at $l$, and the speed $v$ (and thus $m\vec{v}$) remains constant in magnitude if the angular speed $\omega$ is uniform.
Step 4: Analyze the torque and changes to angular momentum
The forces acting on the bob are:
Gravity: $mg$ acting vertically downward
Tension: $T$ in the string directed along the string
The horizontal component of the tension provides the centripetal force, and the vertical component balances the bob's weight. Although the tension and weight produce a net torque about the bob’s center of rotation, the key here is to consider the torque about the point of suspension.
In uniform circular motion, the magnitude of the velocity (and hence linear momentum) remains constant. Consequently, the magnitude of the angular momentum about the pivot also remains constant, because $|\vec{r}|$ does not change and $|\vec{v}|$ is constant.
However, since the bob continually changes its direction of motion around the circle, the direction of $\vec{L} = \vec{r} \times m\vec{v}$ also keeps changing as $\vec{v}$ changes direction. This means $\vec{L}$ rotates with the bob in such a way that its direction changes continuously, but its magnitude does not.
Step 5: Conclude how angular momentum changes
Because the magnitude of angular momentum remains the same but its direction changes steadily during circular motion, the correct statement is:
“Angular momentum changes in direction but not in magnitude.”
