Problem 35 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

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

Step 1: Calculate Force for Potential Energy a) \(U=a y^{3}-b y^{2}\)

First, we need to find the derivative of the potential energy: \(\frac{dU}{dy}\). So, differentiate the given potential energy \(U = a y^{3} - b y^{2}\) with respect to \(y\). \(\frac{dU}{dy} = \frac{d}{dy}(a y^{3} - b y^{2}) = 3 a y^{2} - 2 b y.\) To find the force, we simply multiply this result by \(-1\): \(F(y) = -\frac{dU}{dy} = -(3 a y^{2} - 2 b y) = -3 a y^{2} + 2 b y\).

Step 2: Calculate Force for Potential Energy b) \(U=U_{0} \sin (c y)\)

Again, we need to find the derivative of the potential energy: \(\frac{dU}{dy}\). So, differentiate the given potential energy \(U = U_{0} \sin (c y)\) with respect to \(y\). \(\frac{dU}{dy} = \frac{d}{dy}(U_{0} \sin (c y)) = U_{0} c \cos (c y).\) To find the force, we multiply this result by \(-1\): \(F(y) = -\frac{dU}{dy} = -U_{0} c \cos (c y)\). The force associated with each potential energy is: a) \(F(y) = -3 a y^{2} + 2 b y\) b) \(F(y) = -U_{0} c \cos (c y)\)

Key Concepts

Force Calculation

Calculating force from potential energy is a fundamental concept in physics. When you understand that potential energy can be expressed as a function of position, you can calculate the force exerted by this energy. The force is essentially the negative of the gradient (or the derivative) of potential energy with respect to its position variable.

For example, if you have a potential energy function, say, in terms of the variable position - denoted by \( y \), you find the derivative with respect to \( y \). This derivative gives you the rate of change of potential energy. The resulting derivative represents how the potential energy changes across different positions. By multiplying this derivative by -1, you finally get the force \( F(y) \), as force tends to oppose the direction of increasing potential energy.

Key Points to Remember for Force Calculation from Potential Energy:

• Express potential energy as a function of position \( U(y) \).
• Differentiate \( U(y) \) with respect to \( y \) to find the rate of change.
• Multiply by -1 to determine the force \( F(y) \) that opposes the energy change.

Derivative in Physics

Derivatives play a pivotal role in physics, especially in problems involving motion and forces. A derivative essentially measures how a function changes as its input changes. In the context of physics, derivatives help us understand how physical quantities change over time or position.

Let's consider a simple example in the realm of potential energy. For a potential energy function \( U(y) \), the derivative with respect to \( y \) is \( \frac{dU}{dy} \). This derivative helps to calculate the force exerted by that potential, as it tells us the rate at which the potential energy changes as the position \( y \) changes.

Why Derivatives Matter in Physics:

• They provide the rate of change in quantities like position or energy.
• Derivatives help convert complex physical relationships into more manageable forms.
• They are fundamental in determining motion and understanding forces.

Sinusoidal Potential

Sinusoidal potential refers to potential energy expressed as a sine function. It's a common way to describe oscillating systems where energy changes periodically, such as springs or waves.

A common form is \( U(y) = U_{0} \sin (cy) \), where \( U_{0} \) is the amplitude and \( c \) is a constant that affects the frequency of oscillation. The force derived from a sinusoidal potential, after differentiation and multiplication by -1, is \( F(y) = -U_{0}c \cos (cy) \). The cosine function results naturally because the derivative of sine is cosine.

Features of Sinusoidal Potential in Physical Systems:

• Commonly models periodic or oscillatory systems.
• Easy to compute force due to simple derivatives.
• Useful in studying wave mechanics and harmonic motions.