Question

The potential energy of a certain particle is given by \(U(x, z)=a x^{2}+b z^{3}\), where \(a\) and \(b\) are constants. Find the force vector exerted on the particle.

Short answer

Based on the given potential energy function U(x, z) = ax^2 + bz^3, the force vector exerted on the particle is found to be F(x, z) = (-2ax, -3bz^2). This result was obtained by calculating the gradient of the potential energy function and taking its negative, as per the relationship between potential energy and force.

Step-by-step solution

Calculate the gradient of the potential energy function

To calculate the gradient, we need to find the derivatives of potential energy with respect to x and z. In our function, U(x, z) = ax^2 + bz^3, we will differentiate with respect to x and z and create the gradient vector.

Partial derivative with respect to x

First, let's differentiate the potential energy function with respect to x: \[ \frac{\partial U}{\partial x} = \frac{\partial (ax^2 + bz^3)}{\partial x} = 2ax \]

Partial derivative with respect to z

Now, differentiate the potential energy function with respect to z: \[ \frac{\partial U}{\partial z} = \frac{\partial (ax^2 + bz^3)}{\partial z} = 3bz^2 \]

Gradient of the potential energy function

We can now write the gradient vector using the partial derivatives as its components: \[ \nabla U(x,z) = \left( \frac{\partial U}{\partial x}, \frac{\partial U}{\partial z} \right) = (2ax, 3bz^2) \]

Calculate the force vector

According to the relation between potential energy and force mentioned earlier, the force vector is the negative gradient of the potential energy function. Therefore, we can just negate the gradient vector to obtain the force vector. \[ \vec{F}(x,z) = -\nabla U(x,z) = -(2ax, 3bz^2) = (-2ax, -3bz^2) \]

Conclusion

The force vector exerted on the particle is given by: \[ \vec{F}(x,z) = (-2ax, -3bz^2) \]

Key concepts

Gradient of Potential Energy

When we speak about the gradient of potential energy in physics, we are referring to a vector that points in the direction of the greatest rate of increase of potential energy and whose magnitude is the rate of that increase. In our specific exercise, the gradient is the first step in determining the force acting upon a particle. The potential energy given by the function \(U(x, z) = a x^{2} + b z^{3}\) changes with respect to the positions \(x\) and \(z\), and the gradient allows us to quantify this change in all directions.
To calculate the gradient, we take the derivatives of the potential energy function with respect to each spatial variable \(x\) and \(z\), then assemble these derivatives into a vector. This vector \(abla U(x, z)\), which represents the gradient of potential energy, points in the direction where the particle would experience the maximum increase in potential energy if it were to move infinitesimally in that direction. As a fundamental concept in physics, understanding the gradient not only aids in force calculations but also in visualizing energy landscapes and predicting the motion of particles.

Partial Derivative

A partial derivative is a mathematical operation that shows how a function changes as one of its variables is varied, while keeping other variables constant. In the context of our exercise, the function \(U(x, z)\) depends on two variables, \(x\) and \(z\). To obtain the gradient, we first find the partial derivative of \(U\) with respect to \(x\) while treating \(z\) as a constant, given by \(\frac{\partial U}{\partial x} = 2ax\). Similarly, we also calculate the partial derivative of \(U\) with respect to \(z\), keeping \(x\) constant, yielding \(\frac{\partial U}{\partial z} = 3bz^2\).

The concept of partial derivatives is truly a building block in vector calculus and is vital for understanding how to analyze multidimensional systems. Mastery of partial derivatives enables students to calculate rates of change in systems with multiple variables and is crucial in fields ranging from physics to engineering and economics.

Vector Calculus

Vector calculus is the branch of mathematics that deals with the differentiation and integration of vector fields. It is essential for understanding and describing physical phenomena where variables change in more than one direction. In our exercise, vector calculus is used when we take the partial derivatives of the potential energy function \(U(x, z)\) and create a gradient vector.
This gradient vector, which consists of the partial derivatives, describes how the potential energy changes with position in a multidimensional space. Furthermore, vector calculus is not just limited to calculating gradients; it extends to other operations like divergence, curl, and line integrals, among others. By mastering vector calculus, students can work with forces, electric and magnetic fields, fluid flow, and other vector quantities in a diverse array of scientific and engineering applications.

Force Vector Calculation

The force vector calculation is directly related to Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. In terms of potential energy, force can also be described by considering the negative gradient of the potential energy function. This is due to the fact that the force is directed towards the position of lower potential energy.
In our example, once we have the gradient of \(U(x, z)\), we determine the force vector \(\vec{F}(x,z)\) as the negative of this gradient. Conceptually, this makes sense as the force does 'work' on the particle to move it to a position of lesser energy. Thus, the force vector exerted on the particle is given by \(\vec{F}(x,z) = (-2ax, -3bz^2)\), representing a fundamental relation used to describe the motion and equilibrium of particles in potential fields.