Animated Solution for Physics - Laws of Motion: A ball of mass ($m$) $0.5\text{ kg}$ is attached to the end of a string having length ($l$) $0.5\text{ m}$. The ball is rotated on a horizontal circular path like a physical pendulum about vertical axis. The maximum tension that the string can bear is $324\text{ N}$. The maximum possible value of angular velocity of ball (in rad/s) is

Imagine you are standing in an open field, holding a string with a heavy ball attached to the end. You start spinning it, not vertically like a Ferris wheel, but horizontally, like a lasso. As you spin it faster and faster, the ball rises, tracing out a perfect circle in the air, while the string sweeps out the shape of a cone. This elegant and dynamic system is known in physics as the Conical Pendulum.

It is a classic problem that beautifully marries geometry with Newton's Laws of Motion. But beneath its simple appearance lies a fascinating interplay of forces. In this problem, we are asked to find the absolute maximum angular velocity the ball can achieve before the string, which has a breaking tension of $324$ Newtons, snaps. Let's embark on this journey and decode the physics step by step.

Before we write down any equations, we must understand the physical space the ball occupies. The ball is moving in a horizontal circle. Let's define the length of the string as $l$ and the angle it makes with the vertical axis of rotation as $\theta$.

A common trap students fall into is assuming the radius of the circular path is simply the length of the string $l$. But look closely at the geometry! The string, the vertical axis, and the radius of the horizontal circle form a right-angled triangle.

Using basic trigonometry, the radius $r$ of the horizontal circle is the side opposite to the angle $\theta$. Therefore, we can express the radius as:

This geometric relationship is the crucial first key to unlocking the problem.

Forces in Harmony

Now, let's dive into the dynamics. What forces are actually acting on the ball as it whirls through the air? There are only two real forces at play here. First, the relentless pull of gravity, $mg$, acting straight down. Second, the tension $T$ in the string, pulling the ball along the length of the string towards the pivot point.

Because the tension is acting at an angle $\theta$ to the vertical, it is incredibly helpful to resolve it into two perpendicular components. The vertical component is $T\cos \theta$, pointing upwards. The horizontal component is $T\sin \theta$, pointing directly towards the center of the circular path.

Let's look at the vertical direction first. The ball is moving in a purely horizontal plane; it is neither flying up into the sky nor falling to the ground. This means the vertical forces must be in perfect equilibrium. The upward component of tension perfectly balances the downward pull of gravity:

While this equation is true, it turns out we don't even need it to solve our specific problem! The real magic happens in the horizontal direction.

For any object to move in a circle, it requires a net force directed towards the center of that circle. This is the centripetal force. In our conical pendulum, what is providing this inward pull? It is the horizontal component of the tension!

According to Newton's Second Law for circular motion, the net inward force must equal the mass times the centripetal acceleration. We can write this as:

Here, $\omega$ is the angular velocity we are desperately trying to find. Now, remember that geometric key we found earlier? Let's substitute $r=l\sin \theta$ into our dynamic equation:

Take a moment to appreciate the equation we just derived. Notice something beautiful? As long as the ball is actually spinning (meaning $\theta$ is not zero), we have a $\sin \theta$ term on both sides of the equation.

We can divide both sides by $\sin \theta$, and it completely vanishes from our mathematical reality!

This is a profound physical insight. It tells us that the relationship between the tension in the string and the angular velocity does not depend on the angle $\theta$ at all! The angle adjusts itself naturally, but the core relationship remains pristine and simple.

We are now in the endgame. We want to find the maximum possible angular velocity, ${\omega }_{\max }$. To spin the ball as fast as possible, the string will be stretched to its absolute limit. Therefore, the maximum angular velocity corresponds directly to the maximum tension, $T_{\max }$, the string can bear.

Let's rearrange our simplified equation to solve for $\omega$:

Now, we substitute the maximum tension to find the maximum angular velocity:

It is time to plug in the numbers given in the problem. We know the maximum tension $T_{\max }=324$ N, the mass $m=0.5$ kg, and the length of the string $l=0.5$ m.

Let's compute the denominator first. Multiplying $0.5$ by $0.5$ gives us $0.25$.

Dividing a number by $0.25$ is mathematically identical to multiplying it by $4$. So, we multiply $324$ by $4$, which yields $1296$.

Finally, taking the square root of $1296$ gives us our ultimate answer.

And there we have it! The maximum angular velocity the ball can achieve before the string snaps is $36$ radians per second. A beautiful journey from a physical visualization to a clean, elegant mathematical conclusion.