Problem 719 - JEE Physics

Question

In a gravitational field, at a point where the gravitational potential is zero (A) The gravitational field is necessarily zero (B) The gravitational field is not necessarily zero (C) Nothing can be said definitely, about the gravitational field (D) None of these

Step-by-Step Solution

Answer

In a gravitational field, at a point where the gravitational potential is zero, the gravitational field is the negative derivative of the gravitational potential with respect to distance. As no information about the behavior of gravitational potential around that point is given, nothing can be said definitely about the gravitational field in this situation. Hence, the correct answer is (C) Nothing can be said definitely about the gravitational field.

Step 1: Gravitational Potential and Gravitational Field Formula

In a gravitational field, the gravitational potential (V) is the amount of work done per unit mass to bring an object from infinity to a certain point. The gravitational field (g) represents the force experienced by a unit mass placed at that point. Mathematically, these are represented as: \(V = -\dfrac{GM}{r}\) \(g = \dfrac{GM}{r^2}\) Where \(G\) is the gravitational constant, \(M\) is the mass of the object producing the gravitational field, and \(r\) is the distance from the center of the object.

Step 2: Relationship Between Gravitational Potential and Gravitational Field

Now, we know that the gravitational potential is zero at the given point. We need to find out the behavior of the gravitational field at this point. To do so, let's differentiate the gravitational potential with respect to distance (\(r\)): \(\dfrac{dV}{dr} = \dfrac{d\left(-\dfrac{GM}{r}\right)}{dr} = \dfrac{GM}{r^2} \) \(g = -\dfrac{dV}{dr}\) This equation shows that the gravitational field is the negative derivative of the gravitational potential with respect to distance.

Step 3: Evaluating the Gravitational Field in the Given Situation

According to the given exercise, the gravitational potential is zero at a certain point. It doesn't provide any information about the behavior of gravitational potential around that point. So, we cannot determine if the gravitational field is zero or nonzero at that point. To answer this question, we need more information about the behavior of gravitational potential around that point. Therefore, the correct answer is: (C) Nothing can be said definitely about the gravitational field.

Key Concepts

Gravitational Field

In physics, the gravitational field is a concept that describes the invisible force field around a mass that pulls other objects towards it. It's the force per unit mass, denoted as \( g \), that a small test object would experience at a point in the field.

The formula to calculate gravitational field strength is \( g = \frac{GM}{r^2} \), where:

• \( G \) is the universal gravitational constant (an early discovered constant that plays a critical role in the laws of classical gravitation).
• \( M \) is the mass of the object creating the gravitational field.
• \( r \) is the distance from the center of the mass to the point where the field is being calculated.

Understanding this field is essential because it explains how gravity works in a region around a mass. Even when the gravitational potential (the work needed to move an object into the field) is zero, the actual force may still be present, as the field depends on the derivative of potential with respect to distance.

Gravitational Constant

The gravitational constant, symbolized by \( G \), is a fundamental constant in physics that plays a crucial role in the law of universal gravitation. This constant provides the proportionality factor needed in the gravitational force equation and is measured experimentally.

Its value is approximately \(6.674 \times 10^{-11} \; \text{Nm}^2/\text{kg}^2\). This seemingly small constant shows how relatively weak gravity is compared to other fundamental forces, like electromagnetism.
When used in equations such as \( g = \frac{GM}{r^2} \), the gravitational constant allows us to quantify the force of attraction between two objects. Its discovery was instrumental in helping scientists appreciate that gravity is a universal force that acts consistently throughout the universe, providing a foundation to explain planetary orbits, galactic formations, and more.

Physics Problem-Solving

Solving physics problems, especially those involving gravitational concepts, requires a strategic approach. The process often begins with a clear understanding of the problem statement and identification of the known and unknown variables. Here's a helpful strategy to solve such problems effectively:

• **Understand the Concepts**: Start by grasping the core physics concepts involved, like gravitational potential and fields in this case.
• **Draw Diagrams**: Visual representation can simplify complex situations and reveal hidden relationships between variables.
• **Use Formulas**: Apply relevant equations, such as \( V = -\frac{GM}{r} \) for potential and \( g = \frac{GM}{r^2} \) for the gravitational field, to relate variables.
• **Solve Mathematically**: Isolate the unknowns and simplify the equations step-by-step. Ensure units are consistent to avoid calculation errors.
• **Review the Answer**: Check that the results are physically reasonable and consistent with the problem context.

By setting up problems systematically, you'll not only arrive at the correct answer more often, but also deepen your understanding of how gravitational concepts like potential and fields interrelate.