Problem 455 - JEE Physics

Question

What is the velocity of the bob of a simple pendulum at its mean position, if it is able to rise to vertical height of \(18 \mathrm{~cm}\) (Take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (A) \(0.4 \mathrm{~m} / \mathrm{s}\) (B) \(4 \mathrm{~m} / \mathrm{s}\) (C) \(1.8 \mathrm{~m} / \mathrm{s}\) (D) \(0.6 \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The correct answer is not in the given options. When applying the principle of energy conservation for the given simple pendulum problem, the velocity of the pendulum at its mean position is found to be \(6 \mathrm{~m} / \mathrm{s}\).

Step 1: Understand the Problem

The problem is asking for the velocity of a pendulum bob at its mean position, given the height it achieves when it swings. We will use the principle of conservation of energy to solve it. No matter is given for the bob, so we can ignore it during our calculations because it will be cancelled out.

Step 2: Potential Energy at Maximum Height

The bob's potential energy at its maximum height is given by the formula \(PE = mgh\), where \(PE\) is the potential energy at maximum height, \(m\) is mass of the bob (which we don't need), \(g\) is the acceleration due to gravity and \(h\) is the height. Substituting the given values: \(PE = m * 10 * 0.18\). We don't know the mass of the bob, but we know that it will be the same in our next calculation, so we can keep the formula as it is.

Step 3: Kinetic Energy at Mean Position

The bob's kinetic energy at its mean position is given by the formula \(KE = 0.5 * m * v^2\), where \(KE\) is the kinetic energy, \(m\) is mass (which we don't need), and \(v\) is velocity (which we're trying to find). Since the potential energy at the height is completely converted into kinetic energy at the pendulum's mean position, we can say that \(PE = KE\).

Step 4: Solving for Velocity

Equating the expressions for potential and kinetic energy derived in Step 2 and Step 3 gives us: \(m * 10 * 0.18 = 0.5 * m * v^2\). Notice that 'm' will cancel out. Solving the equation for 'v' (velocity), which is the problem's objective, gives us \(v = \sqrt{2 * 10 * 0.18}\). As a result, the velocity of the bob at the mean position is equal to \(v = 6 m/s\).

Step 5: Conclusion

None of the provided options is correct. The correct solution is \(6 m/s\), which resulted from applying the principle of energy conservation to a simple pendulum problem.

Key Concepts

Conservation of Energy

In a simple pendulum, the principle of conservation of energy plays a crucial role. This principle states that energy cannot be created or destroyed, only transformed from one form to another. As the pendulum swings, it continuously converts between potential energy and kinetic energy.
When the pendulum bob is at its highest point, it has maximum potential energy and zero kinetic energy because it's momentarily at rest. As it swings down to its mean position, the potential energy gradually transforms into kinetic energy. By the time the bob reaches the mean position, all its energy is kinetic. This seamless transformation keeps the total mechanical energy (sum of potential and kinetic) constant, as long as there's no air resistance or friction.
Thus, the initial potential energy at the highest point equals the kinetic energy at the mean position. This is the essence of the pendulum's energy conservation, helping us find the bob's velocity when knowing the height.

Potential Energy

Potential energy in a simple pendulum is primarily due to its height above the lowest point, which we refer to as its mean position. It is calculated using the formula \(PE = mgh\). In this equation, \(m\) represents mass, \(g\) the acceleration due to gravity, and \(h\) the height.
For a pendulum, at its greatest height, the bob holds maximum potential energy because it has been lifted against gravity. When calculating the potential energy, even if we don't know the mass, it doesn't matter in this context. This is because, during the problem-solving process, the mass will cancel out since it's present in both potential and kinetic energy expressions.
The full conversion of this maximum potential energy to kinetic energy as the pendulum swings reaches the mean position is a key aspect of the pendulum's motion and illustrates energy conservation.

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion. In the context of a simple pendulum, this energy is at its peak when the bob is at its mean position. The kinetic energy can be expressed with the formula \(KE = 0.5 \times m \times v^2\), where \(m\) is the mass and \(v\) is the velocity.
At the mean position of the pendulum, all the potential energy from the maximum height has been converted into kinetic energy. This means that the energy due to motion is at its highest here. Solving for velocity at this position involves setting the kinetic energy equal to the initial potential energy and isolating \(v\). In doing this, we can solve for the velocity without requiring the mass of the bob, as it cancels out in the process.
This is a perfect demonstration of how energy transformations empower us to find velocity, even if direct measurements seem impossible.

Mean Position Velocity

The mean position of a pendulum is the point where the pendulum swings through the lowest part of its path. This is where it has the fastest speed because all its energy is kinetic and none is potential anymore.
By conserving energy, we understand that the pendulum's velocity at the mean position can be calculated once we know the height of swing. With potential energy converting entirely to kinetic energy, the equations simplify, removing the need to measure the mass directly. From the formula \(v = \sqrt{2gh}\), derived from energy equality \(PE = KE\), we compute the velocity where \(g\) is gravity's acceleration and \(h\) the maximum height.
In the exercise provided, substituting in the given values yields a velocity of \(6\,\text{m/s}\). While options provided don't include this result, this calculation reinforces the understanding of energy conservation in pendulum motion.