Problem 36 - University Physics with Modern Physics

Question

An object moving in the \(x y\) -plane is acted on by a conservative force described by the potential energy function \(U(x, y)=\) \(\alpha\left(1 / x^{2}+1 / y^{2}\right),\) where \(\alpha\) is a positive constant. Derive an expression for the force expressed in terms of the unit vectors \(\hat{i}\) and \(\hat{y}\) .

Step-by-Step Solution

Answer

The force is \( \vec{F} = 2\alpha \frac{1}{x^3} \hat{i} + 2\alpha \frac{1}{y^3} \hat{j} \).

Step 1: Understand the Relationship between Force and Potential Energy

In classical mechanics, the force associated with a potential energy function can be determined using the gradient of the potential energy. Specifically, the force is given by the negative gradient of the potential energy function. The formula for calculating this is: \[ \vec{F} = -abla U \]where \( abla U \) is the gradient of \( U(x,y) \).

Step 2: Calculate the Gradient of the Potential Energy Function

The potential energy function is given by \( U(x, y) = \alpha \left( \frac{1}{x^2} + \frac{1}{y^2} \right) \). We need to calculate the gradient by taking partial derivatives:\[ \frac{\partial U}{\partial x} = \alpha \frac{d}{dx}\left( \frac{1}{x^2} \right) = -2\alpha \frac{1}{x^3} \]\[ \frac{\partial U}{\partial y} = \alpha \frac{d}{dy}\left( \frac{1}{y^2} \right) = -2\alpha \frac{1}{y^3} \]

Step 3: Express the Gradient as a Vector

The gradient of \( U \), denoted as \( abla U \), can be expressed as:\[ abla U = \left( \frac{\partial U}{\partial x} \right) \hat{i} + \left( \frac{\partial U}{\partial y} \right) \hat{j} = \left( -2\alpha \frac{1}{x^3} \right) \hat{i} + \left( -2\alpha \frac{1}{y^3} \right) \hat{j} \]

Step 4: Compute the Force Using the Negative Gradient

According to Step 1, the force \( \vec{F} \) can be determined as the negative of the gradient of \( U(x,y) \):\[ \vec{F} = -abla U = 2\alpha \frac{1}{x^3} \hat{i} + 2\alpha \frac{1}{y^3} \hat{j} \]This expression represents the force in terms of the unit vectors \( \hat{i} \) and \( \hat{j} \).

Key Concepts

Conservative Force

Conservative forces are unique because they are path-independent. This means the work done by a conservative force when moving an object from one point to another does not depend on the path taken. The only thing that matters is the initial and final positions.

Some common examples of conservative forces include gravity and the electrostatic force.

The key feature of a conservative force is that it can be derived from a potential energy function, denoted as \( U \). This characteristic allows for energy conservation within a system, making it easier to analyze mechanical problems.

When dealing with a conservative force, the total mechanical energy (kinetic plus potential) of a system remains constant, provided no non-conservative forces (like friction) are acting on it. This property simplifies the task of predicting how the system will behave over time.

Gradient of a Function

The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function. It's like a little arrow showing which way to go to climb uphill fastest on a 3D landscape.

In mathematical terms, the gradient is a vector of partial derivatives. For a function of two variables such as \( U(x, y) \), the gradient is given by:

• \( abla U = \left( \frac{\partial U}{\partial x}, \frac{\partial U}{\partial y} \right) \)

This represents how the function \( U \) changes with respect to each variable. For example, \( \frac{\partial U}{\partial x} \) tells us how \( U \) changes as \( x \) changes, holding \( y \) constant. Similarly, \( \frac{\partial U}{\partial y} \) shows the change with \( y \), holding \( x \) constant.

Understanding gradients is crucial because they help us understand how a function behaves and how to use this information, such as determining the direction and magnitude of forces in physics.

Classical Mechanics

Classical mechanics is the branch of physics that deals with the motion of objects and the forces acting upon them. It is based on principles laid down by Isaac Newton, so it's often called Newtonian mechanics. This area of physics covers the motion of everything, from spinning wheels to orbiting planets.

Key concepts in classical mechanics include:

• Newton's Laws of Motion: These laws describe how forces affect motion.
• Work and Energy: The relationship between the force applied and the movement it causes.
• Conservation Laws: These include the conservation of energy and momentum, which are central to predicting physical systems' outcomes.

In problems involving potential energy and forces, classical mechanics uses the concept of energy conservation to make solving problems easier and more intuitive by converting a problem into mathematical language.

Despite its age, classical mechanics remains a powerful tool in science and engineering, offering essential insights for everything from the design of vehicles and structures to the exploration of space.