Problem 1 - Master Resource Book in JEE Main Physics

Question

The bob of a pendulum of mass \(m\) and length \(L\) is displaced, \(90^{\circ}\) from the vertical and gently released. In order that the string may not break upon passing through the lowest point, its minimum strength must be (a) \(m g\) (b) \(2 \mathrm{mg}_{\mathrm{J}}\) (c) \(3 \mathrm{mg}\) (d) \(4 \mathrm{mg}\)

Step-by-Step Solution

Answer

The minimum strength of the string must be 3mg.

Step 1: Identify the forces involved

At the lowest point of the pendulum's swing, the tension in the string and the gravitational force are the primary forces acting on the bob. The tension must be greater than or equal to the gravitational force to prevent breaking.

Step 2: Calculate gravitational force

The gravitational force acting on the bob is calculated as \( F_g = m g \). This force acts downward at all points of the swing.

Step 3: Consider centripetal force at the lowest point

At the lowest point, the bob has maximum kinetic energy and velocity, requiring a centripetal force \( F_c \), which is provided by the tension in the string. The centripetal force can be expressed as \( F_c = \frac{m v^2}{L} \).

Step 4: Apply conservation of energy

When the pendulum is released from the horizontal position, potential energy is converted to kinetic energy at the lowest point. Initially, potential energy \( U = mgh = mgL \). At the lowest point, kinetic energy \( K = \frac{1}{2} m v^2 = mgL \), where \( h = L \) since the pendulum was at 90 degrees.

Step 5: Calculate velocity at the lowest point

Set the gravitational potential energy equal to the kinetic energy at the lowest point: \( mgL = \frac{1}{2} m v^2 \). Solving for \( v^2 \) gives \( v^2 = 2gL \).

Step 6: Find the total tension

The total tension \( T \) in the string at the lowest point is the sum of the gravitational force and the centripetal force, \( T = F_g + F_c = mg + \frac{m v^2}{L} = mg + 2mg = 3mg \).

Step 7: Box the final answer

Compare the calculated tension with the provided options. The minimum strength of the string must be \( 3mg \) to avoid breaking.

Key Concepts

Tension in a Pendulum

When a pendulum swings, especially when it reaches the lowest point, the tension on the pendulum's string is crucial. This tension is the force that keeps the pendulum moving in a circular path. It acts along the string and points toward the pivot point of the pendulum.

The tension is a combination of two forces:

• The gravitational force acting downward.
• The centripetal force needed to keep the pendulum bob moving in a circular path.

At the lowest point, the tension is at its highest because both the gravitational and centripetal forces are acting in the same direction. Hence, the tension in the pendulum is greater than just the gravitational force. In this particular exercise, we calculated that the string needs to endure a tension of at least three times the gravitational force \(3mg\). This ensures the pendulum's string doesn't break.

Conservation of Energy

In the world of physics, the conservation of energy is a fundamental principle. It states that energy cannot be created or destroyed, only converted from one form to another. In a pendulum, energy conversion is a continuous process from potential to kinetic energy and vice versa.

When the pendulum bob is lifted and held at a angle, it possesses maximum potential energy and zero kinetic energy. This potential energy \(mgL\) is due to its height above the lowest point.

Upon release, the potential energy transforms into kinetic energy. At the lowest point, the pendulum has maximum kinetic energy, which can be calculated by setting the initial potential energy equal to the kinetic energy \(\frac{1}{2}mv^2=mgL\). This effective energy transformation is what keeps the pendulum swinging back and forth.

Centripetal Force

Centripetal force is a critical component in pendulum motion. This force is not a separate force acting on the pendulum but is the net force required for circular motion. It acts toward the center of the circle that the pendulum bob traces during its swing.

At the lowest point of the swing, the pendulum bob has the maximum velocity. The centripetal force required at this point ensures that the bob continues in a circular path instead of flying off tangent. It's given by the formula \(F_c = \frac{m v^2}{L}\), where \(v\) is the velocity, \(m\) is mass, and \(L\) is the length of the pendulum.

The need for centripetal force illustrates why the tension is highest at this point. It's this tension (sum of gravitational force and centripetal force) that ensures smooth circular motion.

Gravitational Force Calculation

Gravitational force is the attraction between the pendulum bob and the Earth. It acts constantly downwards throughout the pendulum's motion. Calculating this force is straightforward in pendulum mechanics and provides an essential component of the tension in the system.

In our problem involving a pendulum, the gravitational force is calculated using the simple equation \(F_g = mg\), where \(m\) is the mass of the bob, and \(g\) is the acceleration due to gravity (approximately 9.81 m/s²).

This gravitational force contributes to both the potential energy when the pendulum is elevated and to the net force required for circular motion at all points in its swing. For any pendulum system, understanding and calculating gravitational force is foundational for analyzing its dynamics.