Problem 69 - A Complete Resource Book in Physics for JEE Main

Question

If \(g\) is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass \(m\) raised from the surface of the earth to a height equal to the radius \(R\) of the earth, is (A) \(2 m g R\) (B) \(\frac{1}{2} m g R\) (C) \(\frac{1}{4} m g R\) (D) \(m g R\)

Step-by-Step Solution

Answer

The gain in potential energy of an object of mass \(m\) raised from the surface of the earth to a height equal to the radius \(R\) of the earth is (D) \(mgR\).

Step 1: Identify the given information

We are given the following information: 1. Acceleration due to gravity on Earth's surface, \(g\). 2. Mass of the object, \(m\). 3. Height from Earth's surface to the mass's new position, which is equal to the radius of Earth, \(R\).

Step 2: Set up the potential energy formula

We will use the gravitational potential energy formula: \[PE = mgh\] where \(PE\) is the potential energy, \(m\) is the mass, \(g\) is the acceleration due to gravity, and \(h\) is the height. In this problem, the height, \(h\), is equal to the radius of the Earth, \(R\).

Step 3: Substitute the values into the formula

Now, we will substitute \(g\), \(m\), and \(R\) into the formula: \[PE = mgR\]

Step 4: Find the gain in potential energy

The gain in potential energy in this problem is calculated as: \[ \Delta PE = mgR \]

Step 5: Compare the result with the answer choices

Comparing our result, \(mgR\), with the given answer choices leads us to the correct answer: (D) \(mgR\)

Key Concepts

Acceleration Due to Gravity

When we talk about acceleration due to gravity, we're referring to the constant acceleration that objects experience when they fall towards the Earth or any other celestial body due to its gravitational pull. On Earth, this value is approximately \(9.81 \text{ meters per second squared} (m/s^2)\), commonly denoted as \(g\).

This concept is crucial in understanding how objects move under the influence of Earth's gravity. It applies not just for objects falling directly towards the Earth, but also for those moving away from it, as in the case of our textbook exercise where an object is raised to a certain height. The gravitational force pulling the object back towards the Earth imparts this acceleration, a foundational concept in classical physics.

Work-Energy Theorem

In the realm of physics, the work-energy theorem provides a powerful link between the work done on an object and its kinetic and potential energies. It states that the work done by forces on an object results in a change in its kinetic energy.

Formally, we can express the work-energy theorem as:\[ W = \triangle KE \]
where \(W\) is the work done and \(\triangle KE\) is the change in kinetic energy. However, this concept extends to potential energy in cases where gravity does the work. When an object is raised against Earth's gravity, the work done by the applied force is stored as gravitational potential energy. So, when students struggle to connect work with potential energy, it's essential to explain that in lifting an object, work is done against the gravitational force, resulting in an increase in the object's potential energy, as seen in our exercise.

Gravitational Force

Lastly, let's delve into the concept of gravitational force. It's the attractive force between two masses, such as the Earth and an object on its surface. Isaac Newton described this force as proportional to the product of the two masses and inversely proportional to the square of the distance between their centers.

For objects near Earth's surface, we can simplify the universal law of gravitation, leading to the equation for weight (\(F = mg\)), where \(F\) is the gravitational force, \(m\) is the mass of the object, and \(g\) is the acceleration due to gravity. In our textbook exercise, this gravitational force is what would bring the object back to Earth if it were to fall, converting its potential energy back into kinetic energy until it comes to a rest upon impact, fully demonstrating the interconnectedness of these fundamental physics concepts.