Question

A simple pendulum having length \(\ell\) is suspended at the roof of a train moving with constant acceleration 'a' along horizontal direction. The periodic time of this pendulum is.... (A) \(\mathrm{T}=2 \pi \sqrt{(\ell / \mathrm{g})}\) (B) \(\mathrm{T}=2 \pi \sqrt{\\{\ell /(\mathrm{g}+\mathrm{a})\\}}\) (C) \(\mathrm{T}=2 \pi \sqrt{\\{\ell /(\mathrm{g}-\mathrm{a})\\}}\) (D) \(\left.\mathrm{T}=2 \pi \sqrt{\\{\ell} /\left(\mathrm{g}^{2}+\mathrm{a}^{2}\right)\right\\}\)

Short answer

The short answer based on the given step-by-step solution is: (D) \(\mathrm{T}=2 \pi \sqrt{\\{\ell}/\left(\mathrm{g}^{2}+\mathrm{a}^{2}\right)\\}\)

Step-by-step solution

Analyze the Situation in the Non-Inertial Frame of Reference

In the train's frame of reference (which is not an inertial frame due to acceleration 'a'), there is a pseudo force acting on the pendulum opposite to the direction of the train's acceleration. This force is equal to \(-m\mathrm{a}\) where m is the mass of the pendulum bob.

Determine the Effective Gravity Acting on the Pendulum

In addition to the actual gravitational force \(\downarrow m\mathrm{g}\) acting on the pendulum, we must consider the pseudo force \(\leftarrow (-m\mathrm{a})\) acting in the train's frame of reference. These combined forces form a resultant force making an angle with the vertical direction. The effective gravity 'g_eff' acting on the pendulum is the total force divided by the mass of the bob: \( g_{\mathrm{eff}} = \sqrt{g^{2} + a^{2}} \)

Apply the Simple Pendulum Formula

Now that we know the effective gravity acting on the pendulum, we can apply the formula for the period of a simple pendulum: \( \mathrm{T} = 2\pi\sqrt{\frac{\ell}{g_{\mathrm{eff}}}} \)

Substitute the Effective Gravity into the Formula

Replace g_eff in the formula with the expression found in Step 2: \( \mathrm{T} = 2\pi\sqrt{\frac{\ell}{\sqrt{g^{2} + a^{2}}}} \) This matches the Option (D), so the correct answer is: (D) \(\mathrm{T}=2 \pi \sqrt{\\{\ell}/\left(\mathrm{g}^{2}+\mathrm{a}^{2}\right)\\}\)

Key concepts

Non-Inertial Frame of Reference

When dealing with motion in a non-inertial frame of reference, we consider environments that are accelerating. In contrast to inertial frames, where Newton's laws appear straightforward, a non-inertial frame like a train accelerating with constant acceleration 'a' leads to apparent extra forces. These are not real forces but need to be accounted for when analyzing motion and dynamics from within the frame. The pendulum inside this train experiences such conditions. Because the train is accelerating, it creates a situation where the normal rules of physics within that frame look unusual. This acceleration causes the effect known as pseudo force, which impacts the pendulum's motion. Understanding a non-inertial frame is crucial because it modifies how forces interact within that system and ultimately how the pendulum behaves.

Effective Gravity

The concept of effective gravity is key to understanding motion in non-inertial frames. Effective gravity is essentially what we perceive as the force of gravity, combined with other forces acting on a body. In the case of our pendulum inside a moving train, the pendulum is affected not only by the Earth's gravitational pull but also by the pseudo force resulting from the train's acceleration. These combined forces can be thought of as a single "effective" gravitational force. To calculate it, we merge the actual gravitational force and the pseudo force into one resultant force. This force makes an angle with the vertical and has a magnitude given by \[ g_{\text{eff}} = \sqrt{g^2 + a^2} \]This effective gravity allows us to apply the physics of pendulum motion accurately while accounting for the unusual conditions. It ensures that what might seem like complex motion remains predictable and understandable.

Pseudo Force

A pseudo force, often called a fictitious force, appears when observing motion from an accelerating frame of reference. It does not arise from any physical interaction but is a result of the acceleration of the frame itself. In the problem with the pendulum, the pseudo force acts horizontally, opposite to the train's direction of acceleration. It has a magnitude of \[ -m \cdot a \]where 'm' is the mass of the pendulum. This pseudo force alters the apparent gravitational force experienced by the pendulum. While it is a mere artifact of observing from a non-inertial frame, its effect is very real in terms of how the pendulum swings. Acknowledging pseudo forces is critical when working through problems within accelerating systems, providing the correct framework to analyze motion.

Period of Pendulum

The period of a pendulum is the time it takes for the pendulum to complete one full swing, back and forth. In situations like a pendulum in an accelerating frame, calculating the period involves understanding how both gravity and pseudo forces affect the pendulum. Normally, the formula for the period of a simple pendulum is \[ T = 2\pi \sqrt{\frac{\ell}{g}} \]where \(\ell\) is the length of the pendulum and \(g\) is the acceleration due to gravity. However, in the presence of effective gravity \(g_{\text{eff}} = \sqrt{g^2 + a^2}\) resulting from the addition of a pseudo force, the formula becomes \[ T = 2\pi \sqrt{\frac{\ell}{g_{\text{eff}}}} \]This modified period accounts for the changes due to acceleration and is vital for accurately describing the pendulum's motion in non-standard environments. It shows how fundamental forces and observed mechanics interact intricately, helping us predict motion under complex conditions.