Question

Calculate the force \(F(y)\) associated with each of the following potential energies: a) \(U(y)=a y^{3}-b y^{2}\) b) \(U(y)=U_{0} \sin (c y)\)

Short answer

Question: Calculate the force associated with the given potential energies. a) Potential energy: \(U(y)=a y^{3} - b y^{2}\) b) Potential energy: \(U(y)=U_{0} \sin (c y)\) Answer: a) The force associated with the potential energy \(U(y)=a y^{3} - b y^{2}\) is: $$F(y) = -3a y^{2} + 2b y$$ b) The force associated with the potential energy \(U(y)=U_{0} \sin (c y)\) is: $$F(y) = -U_{0} c \cos (c y)$$

Step-by-step solution

a) Potential energy: \(U(y)=a y^{3} - b y^{2}\)

First, differentiate \(U(y)\) with respect to \(y\): $$\frac{dU}{dy} = 3a y^{2} - 2b y$$ Now calculate the force \(F(y)\): $$F(y) = -\frac{dU}{dy} = -(3a y^{2} - 2b y) = -3a y^{2} + 2b y$$ The force associated with this potential energy is: $$F(y) = -3a y^{2} + 2b y$$

b) Potential energy: \(U(y)=U_{0} \sin (c y)\)

First, differentiate \(U(y)\) with respect to \(y\): $$\frac{dU}{dy} = U_{0} \cos (c y) \cdot c$$ Now calculate the force \(F(y)\): $$F(y) = -\frac{dU}{dy} = -U_{0} \cos (c y) \cdot c$$ The force associated with this potential energy is: $$F(y) = -U_{0} c \cos (c y)$$

Key concepts

Physics Potential Energy

In the world of physics, potential energy is the energy held by an object due to its position relative to other objects, stress within itself, its electric charge, or other factors. For example, a ball held at a certain height above the ground has gravitational potential energy due to its elevated position in Earth's gravitational field.

There are various forms of potential energy, each associated with a particular type of force. Gravitational potential energy is related to the gravitational force, elastic potential energy with elastic force, and so on. The important aspect of potential energy is that it has the potential to be converted into kinetic energy—the energy of motion—as the object's state changes.

Understanding potential energy is crucial for solving problems in mechanics, where the total energy (kinetic plus potential) is often conserved. By analyzing potential energy, we can predict how an object might move when allowed to act under the influence of a conservative force, such as gravity.

Differentiation in Physics

The concept of differentiation is a fundamental tool in physics, especially when analyzing motion and energy. Differentiation, in mathematical terms, is the process of finding the derivative of a function, which represents the rate at which a quantity changes with respect to another quantity.

In physics, we use differentiation to find rates like velocity (the rate of change of position with time) and acceleration (the rate of change of velocity with time). When dealing with potential energy, the derivative of the potential energy function with respect to position gives us the force exerted by the potential energy on an object.

This process is key to solving various physics problems. In the context of the exercise, by differentiating the potential energy function with respect to the variable 'y', we are effectively analyzing how the force changes as the object’s position changes within a particular field, whether it be a gravitational field, an elastic medium, or any other field described by a potential energy function.

Force Associated with Potential Energy

The link between force and potential energy is given by the negative derivative of the potential energy with respect to position. This relationship is a cornerstone of classical mechanics and is encapsulated in conservative forces—forces that have the property where the work done in moving an object between two points is independent of the path taken.

Mathematically, if we have a potential energy function, say, U(y), then the force F(y) associated with that potential energy is given by the negative gradient of U(y). This means we can write:
\[ F(y) = -\frac{dU}{dy} \]
By applying this concept to the given problems in the exercise, the force associated with each potential energy function can be calculated through differentiation. This relationship is critical for understanding how objects will move within a field or under a given set of forces and is instrumental for students in fields from engineering to environmental science.