Question

Some forces in nature vary with the inverse of the distance squared between two objects. For a force like this, how does the potential energy vary with the distance between the two objects? a) The potential energy varies with the distance. b) The potential energy varies with the distance squared. c) The potential energy varies with the inverse of the distance. d) The potential energy varies with the inverse of the distance squared. e) The potential energy does not depend on the distance.

Short answer

Answer: c) The potential energy varies with the inverse of the distance.

Step-by-step solution

Recall the relationship between force and potential energy

To determine how potential energy varies with distance, it's important to recall the relationship between force and potential energy. The work done by a force, when moving an object from point A to point B, is equal to the change in potential energy (∆U) of the object.

Write down the given force function

The problem states that the force varies with the inverse of the distance squared between the two objects. Mathematically, this can be represented as: F(r) ∝ 1/r^2, where F(r) is the force and r is the distance between the two objects.

Calculate work done by the force

To find the change in potential energy, we need to find the work done by the force when the distance changes. We can do this by integrating the force function with respect to distance (r) over the given limits. Work done or ∆U = ∫ F(r) dr = ∫ (k / r^2) dr, where k is a constant of proportionality.

Integrate the force function with respect to distance

Now, let's integrate the force function to find the potential energy change. ∆U = ∫ (k / r^2) dr = - k * ∫ (1/r^2) dr = -k * (-1/r) = k/r + C, where C is the integration constant. Thus, potential energy U(r) is given as U(r) = k/r + C.

Identify the variation of potential energy with distance

Now that we found the potential energy function, we can identify how it varies with distance: U(r) = k/r + C We can see that the potential energy varies with the inverse of the distance (r) between the two objects. The correct answer is: c) The potential energy varies with the inverse of the distance.

Key concepts

Potential Energy Variation

The concept of potential energy variation in physics ties closely with the way forces behave over a distance. When learning about forces that follow the inverse square law, such as gravity or electrostatic forces, it's crucial to understand how this affects potential energy. As distance increases, the strength of these forces decreases sharply, following a pattern of being inversely proportional to the square of the distance, that is \( \frac{1}{r^2} \).

In practical terms, if you were to move an object in the field of such a force, its potential energy would not diminish at a constant rate. Instead, as the object gets farther away, every additional unit of distance results in a smaller change in potential energy compared to the previous unit. This relationship is vital to fields like astrophysics and electrical engineering, where understanding these variations can mean the difference between a satellite remaining in stable orbit or drifting off into space.

Force and Potential Energy Relationship

The relationship between force and potential energy is both fundamental and profoundly insightful in physics. A force applied over a certain distance does work, and this work is realized as a change in potential energy. For inverse square forces, this relationship becomes particularly interesting.

Imagine pushing a block up a hill; the work you do against gravity is stored as gravitational potential energy. Similarly, when dealing with forces that diminish with the square of the distance, the work done in moving an object away increases its potential energy relative to the force source. However, because the force weakens with distance, the amount of work required to move the object further away decreases exponentially, thus increasing its potential energy at a rate inversely proportional to the distance.

Work Done by Force

To fully grasp the concept of work done by force, it's essential to link it to real-world actions. Work is all about applying a force over a distance. When you lift a book, carry a bag, or pull a wagon, you're doing work by exerting a force to move an object against another force (like gravity, friction, etc.).

In the context of the inverse square law, work done reflects the energy required to move an object within a force field that changes intensity with distance. The work calculation involves integrating the force over a path, and as seen in the textbook exercise, it's the integral of force that unveils how potential energy changes. Grounding this concept in familiar experiences helps illuminate the abstractness and makes it more accessible for students to understand.

Integration in Physics

Integration is a powerful tool in physics for dealing with a wide variety of problems, particularly when phenomena change continuously. It allows physicists to sum up infinitely small quantities to uncover the total effect, like the work done by a varying force.

Think of integration like adding up an uncountable number of very thin slices to find the volume of a loaf of bread. Applying this to forces, when you integrate a force over a distance, you're effectively summing the work done across every infinitesimal segment between two points. This calculation is not just a mathematical exercise but a reflection of physical processes, tying together concepts, enhancing understanding, and revealing how physics governs everything from the orbit of planets to the flow of electricity in a wire.