Problem 23 - Master Resource Book in JEE Main Physics

Question

At a given place where, acceleration due to gravity is \(g \mathrm{~ms}^{-2}\), a sphere of lead of density \(d \mathrm{kgm}^{-3}\) is gently released in a column of liquid of density \(\rho \mathrm{kgm}^{-3}\). If \(d>\rho\), the sphere will (a) fall vertically with an acceleration of \(\mathrm{g} \mathrm{ms}^{-2}\) (b) fall vertically with no acceleration (c) fall vertically with an acceleration \(g\left(\frac{d-\rho}{d}\right)\) (d) fall vertically with an acceleration \(\rho / d\)

Step-by-Step Solution

Answer

Option (c): The sphere falls with an acceleration \( g \left(\frac{d-\rho}{d}\right) \).

Step 1: Understand Buoyancy and Acceleration

When the lead sphere is released in the liquid, two main forces act on it: the gravitational force pulling it downwards and the buoyant force pushing it upwards. According to Archimedes' principle, the buoyant force is equal to the weight of the liquid displaced by the sphere. The net force acting on the sphere therefore determines its acceleration.

Step 2: Calculate the Gravitational Force

The gravitational force acting on the sphere is given by \( F_{ ext{gravity}} = V imes d imes g \), where \( V \) is the volume of the sphere, \( d \) is the density of the sphere, and \( g \) is the acceleration due to gravity.

Step 3: Calculate the Buoyant Force

The buoyant force is given by \( F_{ ext{buoyant}} = V imes \rho imes g \), where \( V \) is the volume of the sphere and \( \rho \) is the density of the liquid. This force acts in the upward direction.

Step 4: Determine the Net Force

The net force acting on the sphere is the difference between the gravitational force and the buoyant force. This can be expressed as: \( F_{ ext{net}} = V imes d imes g - V \times \rho imes g = V imes g \times (d - \rho) \).

Step 5: Apply Newton's Second Law

According to Newton's second law, the acceleration \( a \) of the sphere is given by the net force divided by the mass of the sphere: \( a = \frac{F_{ ext{net}}}{V imes d} = \frac{V imes g \times (d - \rho)}{V imes d} = g \times \left(\frac{d - \rho}{d}\right) \).

Step 6: Conclude with the Correct Option

From the calculated expression for acceleration, the correct answer is option (c): the sphere will fall vertically with an acceleration \( g \times \left(\frac{d - \rho}{d}\right) \).

Key Concepts

Acceleration due to Gravity

The concept of acceleration due to gravity is fundamental in understanding the motion of objects on Earth. It is denoted by the symbol \( g \) and has an approximate value of \( 9.8 \, \text{m/s}^2 \) near the Earth's surface. This constant acceleration affects how objects move towards the planet. It applies to any freely falling object, meaning that the object is moving solely under the influence of gravity.

Key points to remember about acceleration due to gravity include:

• It acts downward, towards the center of the Earth.
• The magnitude of \( g \) can vary slightly based on location due to Earth's shape and rotation.
• It is responsible for the sensation of weight.

All objects, regardless of their mass, experience the same acceleration due to gravity when in freefall. This concept is critical for understanding how the lead sphere in the exercise behaves differently when submerged in a liquid as compared to air.

Archimedes' Principle

Archimedes' Principle is a key concept in fluid mechanics that helps us understand how objects behave when submerged in a fluid. According to this principle, "any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object." This principle is named after the ancient Greek scientist Archimedes. It helps explain why objects float or sink.

Some core ideas from Archimedes' Principle include:

• Buoyant force is directed upward, opposing the force of gravity.
• The magnitude of the buoyant force equals the weight of the displaced fluid.
• If the buoyant force is greater than the weight of the object, the object will float.
• If the buoyant force is less than the weight of the object, the object will sink.

In the exercise, this principle explains the buoyant force acting on the lead sphere when it's submerged in liquid. Since the sphere's density is greater than the liquid's, it will not float; instead, it sinks, but its fall is mitigated by the upward buoyant force.

Newton's Second Law

Newton's Second Law of Motion provides a formula that directly connects the forces acting on an object and its resulting motion. The law states, "the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass." This can be mathematically expressed as \( F = ma \), where \( F \) is the net force, \( m \) is the mass, and \( a \) is the acceleration.

Key elements of Newton's Second Law include:

• Acceleration is caused by net forces acting on an object.
• If the net force is zero, the object's velocity remains constant.
• Increased force results in increased acceleration given the same mass.

Incorporating the heavy concepts of buoyancy and gravity, the law allows us to determine the lead sphere's acceleration when submerged in a liquid. It considers the sphere's mass and the net force, which results from opposing gravitational and buoyant forces, explaining why the sphere in the exercise falls at the calculated acceleration \( g \left( \frac{d-\rho}{d} \right) \). This highlights the interaction between mass, gravity, and buoyancy according to Newton's framework.