Animated Solution for Physics - Laws of Motion: A simple pendulum of length $L$ and mass (bob) $M$ is oscillating in a plane about a vertical line between angular limits $-\varphi$ and $+\varphi$. For an angular displacement $\theta$ ($|\theta |<\varphi$), the tension in the string and the velocity of the bob are $T$ and $v$ respectively. The following relations hold good under the above conditions.

When we look at a simple pendulum swinging back and forth, it is easy to get lost in the hypnotic rhythm and forget the rigorous physics governing its motion. A pendulum is not just a mass on a string; it is an object executing non-uniform circular motion in a vertical plane.

Imagine the bob of mass $M$ suspended by a string of length $L$. As it swings, it traces out a perfect circular arc. At any arbitrary angle $\theta$ from the vertical, the bob possesses a certain instantaneous velocity $v$. To truly understand the dynamics of this system, we must look beyond the obvious and dissect the forces at play.

In standard kinematics, we often default to resolving forces along the horizontal ($x$) and vertical ($y$) axes. However, in circular motion, this approach leads to a mathematical nightmare. Why? Because the acceleration of the bob is constantly changing direction as it moves along the curve.

Instead, we deploy a much more elegant weapon: the Radial and Tangential coordinate system. We define one axis along the string (the radial direction) and the other perpendicular to it (the tangential direction). This choice is brilliant because it perfectly aligns with the two distinct types of acceleration the bob experiences: centripetal acceleration (which changes the direction of velocity) and tangential acceleration (which changes the magnitude of velocity).

The Tug of War

Let's analyze the forces along the radial direction. The string pulls the bob towards the pivot with a Tension $T$. Simultaneously, gravity pulls the bob straight down with a force $Mg$.

Because gravity is vertical, it is not aligned with our new axes. We must resolve it. By simple geometry, the angle between the downward gravity vector and the extended radial line is also $\theta$. Therefore, the component of gravity pulling the bob away from the pivot is $Mg\cos \theta$.

Now, according to Newton's Second Law for circular motion, an object requires a net inward force—the centripetal force—to stay on its circular path. This force is given by $\frac{M{v}^2}{L}$.

The net inward force is the inward Tension minus the outward gravity component. Setting up our master equation, we get:

This beautiful equation tells us that the tension must not only support the outward pull of gravity but also provide the necessary centripetal force to keep the bob turning. This perfectly validates option (b).

The Restoring Force

What about the tangential direction? The tension $T$ acts purely radially, so it has zero component along the tangent. The only force acting tangentially is the other component of gravity: $Mg\sin \theta$.

This force points towards the lowest point of the swing (the mean position). It is the 'restoring force' that constantly tries to bring the pendulum back to equilibrium.

Applying Newton's Second Law along the tangent, the net tangential force must equal mass times tangential acceleration ($a_T$):

By elegantly canceling the mass $M$ from both sides, we arrive at the tangential acceleration:

This confirms that option (c) is absolutely correct. The tangential acceleration is independent of the mass and is maximum at the extreme positions where $\theta$ is largest.

Through careful resolution of forces, we have unveiled the true dynamics of the simple pendulum. Option (a), $T\cos \theta =Mg$, is a trap meant for students confusing this with a conical pendulum. Option (d), $T=Mg\cos \theta$, is a static illusion, true only at the exact moment the bob stops at its extreme positions ($v=0$).

By mastering the radial and tangential axes, you don't just solve a problem; you gain a universal tool for conquering any vertical circular motion challenge in physics.