Question

The bob of a simple pendulum having length ' \(\ell\) ' is displaced from its equilibrium position by an angle of \(\theta\) and released. If the velocity of the bob, while passing through its equilibrium position is \(\mathrm{v}\), then \(\mathrm{v}=\ldots \ldots \ldots\) (A) \(\sqrt{\\{2 g \ell(1-\cos \theta)\\}}\) (B) \(\sqrt{\\{2 g \ell(1+\sin \theta)\\}}\) (C) \(\sqrt{\\{2 g \ell(1-\sin \theta)\\}}\) (D) \(\sqrt{\\{2 g \ell(1+\cos \theta)\\}}\)

Short answer

The short answer is: \(v = \sqrt{2gl(1-\cos \theta)}\).

Step-by-step solution

STEP 1: Identify the initial and final points

In this situation, we will consider two points: - Initial point: when the pendulum is released from an angle of θ with respect to vertical. - Final point: when the pendulum passes through its equilibrium position (vertical).

STEP 2: Calculate the initial potential and kinetic energy

Initially, the bob is at an angle θ and at rest (with an initial velocity of 0). So, the initial kinetic energy is zero. We need to find the initial potential energy. As the bob is raised by a vertical height of \((l-l\cos \theta)\), the initial potential energy is: \[PE_{i}=mgh=mgl(1-\cos \theta)\]

STEP 3: Calculate the final potential and kinetic energy

As the pendulum passes through its equilibrium position, its gravitational potential energy becomes zero. We need to find the final kinetic energy, which is given by: \[ KE_{f} = \frac{1}{2}mv^2\]

STEP 4: Use the conservation of energy principle

According to the conservation of mechanical energy: \[ PE_{i} + KE_{i} = PE_{f} + KE_{f}\] Since \(PE_{i} = mgl(1-\cos \theta)\), \(KE_{i} = 0\), \(PE_{f} = 0\), and \(KE_{f} = \frac{1}{2}mv^2\), the equation becomes: \[mgl(1-\cos \theta) = \frac{1}{2}mv^2\]

STEP 5: Solve the equation and find the velocity

Now, we need to solve the equation for the velocity 'v': \[v^2 = 2gl(1-\cos \theta)\] Taking the square root of both sides, we get: \[v = \sqrt{2gl(1-\cos \theta)}\] Comparing the expression for velocity with the given options, we can conclude that the correct answer is: (A) \(v = \sqrt{\\{2 g \ell(1-\cos \theta)\\}}\)

Key concepts

Conservation of Energy

In physics, conservation of energy is a fundamental principle stating that energy cannot be created or destroyed; it can only be transferred or converted from one form to another. This is a key concept in understanding how a simple pendulum works.

When the pendulum bob is lifted to a height, it has potential energy because of its position. As the bob swings downwards towards the equilibrium position, the potential energy is converted into kinetic energy.

When using the conservation of energy principle:

• The total energy at the start (potential energy) must equal the total energy at the energetic point (kinetic energy) as the pendulum swings down.
• This principle helps us find the velocity of the bob when it passes through its equilibrium position. By equating the initial potential energy with the final kinetic energy, we can derive expressions for various quantities such as speed.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy that an object possesses due to its position in a gravitational field. For the pendulum, GPE is highest when the pendulum is at its maximum displacement because it is at the highest point of its swing.

The formula for GPE in the context of a pendulum is:

• \[ PE_i = mgh = mgl(1-\cos \theta) \]

Here,

• \( m \) represents the mass of the pendulum bob.
• \( g \) is the acceleration due to gravity.
• \( l(1-\cos \theta) \) is the vertical height the bob is lifted.

At the equilibrium position (the lowest point), GPE is zero because the height is zero, showing that all potential energy has been converted to kinetic energy.

Kinetic Energy

Kinetic energy (KE) is the energy of an object due to its motion. For the pendulum, kinetic energy is zero at the highest point when the bob is momentarily at rest before it begins its descent. However, at the equilibrium position, it's at its maximum because the velocity of the bob is greatest.

The formula for kinetic energy is:

• \[ KE_f = \frac{1}{2}mv^2 \]

In this case,

• \( KE_f \) is calculated when the pendulum is at the equilibrium position.
• \( v \) represents the speed of the bob as it swings through this position.

The conservation of energy tells us that all the initial potential energy has been converted into kinetic energy as the pendulum reaches the equilibrium position.

Equilibrium Position

The equilibrium position is significant for the pendulum as it represents the point where the gravitational forces and the restoring forces are balanced.

When the pendulum swings, it passes through this vertical alignment with the maximum speed and zero potential energy. It is a pivotal point in its motion:

• The speed of the pendulum is the highest at this point.
• All energy has shifted from potential to kinetic.
• Calculating speed at this position helps understand the energy dynamics of pendular motion.

Understanding the equilibrium position allows us to apply the conservation of energy principle effectively to compute other values like velocity, using derived equations such as:

\[ v = \sqrt{2gl(1-\cos \theta)} \]