Problem 14 - Master Resource Book in JEE Main Physics

Question

The string of a pendulum of length \(l\) is displaced through \(90^{\circ}\) from the vertical and released. Then, the minimum strength of the string in order to withstand the tension as the pendulum passes through the mean position is (a) \(\overline{m g}\) (b) \(6 \mathrm{mg}\) (c) \(3 \mathrm{mg}\) (d) \(5 \mathrm{mg}\)

Step-by-Step Solution

Answer

The minimum strength of the string is 3mg, option (c).

Step 1: Understanding the Problem

We are given a pendulum with a string of length \(l\) that is displaced to \(90^{\circ}\) from the vertical position and then released. As the pendulum swings, it is essential to determine the minimum tension the string can withstand when the pendulum is at its lowest point (the mean position). This tension will include gravitational forces and centripetal forces due to the pendulum's motion.

Step 2: Calculate the Potential Energy at the Starting Point

At \(90^{\circ}\), the pendulum is at its highest point. The initial potential energy (PE) is given by \(PE = mgh\). Since the height \(h = l\), the potential energy is \(PE = mgl\). This is because the entire length of the string forms the height from which the bob swings down.

Step 3: Determine the Kinetic Energy at the Mean Position

At the mean position, all the potential energy has been converted into kinetic energy (KE), as the height relative to the lowest point is zero. So, \(KE = \frac{1}{2}mv^2 = mgl\).

Step 4: Solve for Velocity at the Mean Position

From the kinetic energy equation \(\frac{1}{2}mv^2 = mgl\), solve for velocity \(v\): \[v = \sqrt{2gl}\]

Step 5: Calculate Tension at the Mean Position

At the mean position, the tension \(T\) in the string is the sum of the gravitational force \(mg\) and the centripetal force \(\frac{mv^2}{l}\) required to keep the bob moving in a circular path: \[T = mg + \frac{mv^2}{l}\]

Step 6: Substitute for Velocity and Simplify

Substitute the expression for \(v\) from the previous step into the tension equation: \[T = mg + \frac{m(2gl)}{l}\] \[T = mg + 2mg\] Thus, \(T = 3mg\).

Step 7: Conclusion

The minimum strength of the string, which corresponds to the tension at the lowest point, needs to be \(3mg\) for the pendulum to swing properly without breaking. This matches option (c).

Key Concepts

Simple Pendulum

A simple pendulum consists of a mass, called the bob, suspended from a fixed point by a string of length \(l\) that can swing freely in a plane. When the pendulum is displaced from its rest position (the vertical), it is free to swing back and forth under the influence of gravity. This type of pendulum assumes no air resistance or friction at the pivot, allowing for a clear demonstration of basic physical principles.

• Motion depends primarily on gravitational forces acting on the mass.
• Typically observed as a model system in classrooms to demonstrate harmonic motion.
• The length of the string and gravity are the main factors determining the period of the pendulum.

The pendulum swings through the vertical by converting potential energy to kinetic energy, demonstrating the conservation of mechanical energy in its simplest form.

Potential Energy

Potential energy describes the stored energy in an object due to its position relative to a reference point, often the ground. In a pendulum, the potential energy is maximized when the pendulum is at its highest displacement point.

• For a pendulum, potential energy is calculated using the formula \(PE = mgh\), where \(m\) is mass, \(g\) is the gravitational constant, and \(h\) is the height.
• In the given problem, the pendulum is displaced to \(90^{\circ}\), so the entire length of the string \(l\) determines the height \(h = l\).
• The potential energy gets converted to kinetic energy as the pendulum swings down.

Understanding potential energy helps us determine how much energy is available to be converted into motion as the pendulum swings.

Kinetic Energy

Kinetic energy is the energy of motion, present in the pendulum bob as it swings through the lowest point of its path (mean position). As potential energy is converted to kinetic energy, the pendulum experiences its maximum velocity.

• At the mean position, potential energy is zero, and kinetic energy is maximum.
• Calculated using \(KE = \frac{1}{2}mv^2\), reflecting the velocity of the moving mass.
• In this scenario, the entire initial potential energy \(mgl\) is converted into kinetic energy, so \(KE = mgl\).

By converting potential energy to kinetic energy at the mean position, we calculate the velocity and necessary tension in the string to keep the pendulum moving without breaking the string.

Centripetal Force

Centripetal force is the force required to keep a moving object traveling in a circular path. For pendulums, this force is crucial at the mean position where the velocity is highest.

• Centripetal force acts inward, pulling the mass towards the center of its circular path.
• Calculated using \(\frac{mv^2}{l}\), where \(v\) is the velocity and \(l\) is the length of the pendulum.
• Essential for determining the tension the string must withstand at the lowest point.

In our pendulum example, the centripetal force combines with the gravitational force to form the total tension \(T = mg + \frac{mv^2}{l}\), emphasizing its role in keeping the pendulum under tension without snapping the string.