Animated Solution for Physics - Rotational Motion: Comprehension Passage

A Sliding Bob

Imagine a pendulum bob resting peacefully on a frictionless floor. Suddenly, it receives a sharp horizontal impulse, $P=0.2\text{ kg-m/s}$. This impulse instantly imparts a horizontal velocity to the bob. Since the mass of the bob is $m=0.1\text{ kg}$, we can easily find this initial velocity:

The bob begins to slide horizontally across the floor. At this stage, the string is slack, and the bob moves in a straight line.

We are asked to find the angular momentum of the bob about the suspension point just before the string becomes taut. The definition of angular momentum is the cross product of the position vector and the linear momentum vector, $\vec{J}=\vec{r}\times \vec{p}$.

Because the linear momentum is purely horizontal, the perpendicular distance from the suspension point to the line of motion is simply the height of the suspension point above the floor, $H=0.9\text{ m}$. Therefore, the magnitude of the angular momentum is:

Substituting the given values, we get:

This gives us the answer to the first part of the problem.

The String Becomes Taut

As the bob slides further away, the distance from the suspension point increases. The critical moment arrives when this distance exactly equals the length of the string, $L=1.0\text{ m}$. At this instant, the string becomes taut. Let's define the angle the string makes with the vertical at this moment as $\theta$. Using simple right-triangle geometry, we can find the cosine of this angle:

Here is the crucial physics catch: When the inextensible string suddenly jerks taut, it exerts an impulsive tension along its length. This impulsive force completely destroys the component of the bob's velocity that is parallel to the string ($v_{\parallel }$). However, because there is no impulsive force perpendicular to the string, the perpendicular component of the velocity ($v_{\perp }$) is perfectly conserved.

We must resolve the initial horizontal velocity $v_0$ into these two components. By geometry, the angle between the horizontal velocity vector and the direction perpendicular to the string is also $\theta$. Therefore, the surviving velocity is:

Finally, we can calculate the kinetic energy of the pendulum just after lift-off. It is simply the kinetic energy associated with this surviving perpendicular velocity:

The official answer key provides $0.16$ as the answer, which is a slight truncation of our exact result. Understanding the mechanics of the impulsive jerk is the key to mastering this beautiful problem!