Question

Gravitational potential The gravitational force between two point masses \(M\) and \(m\) is $$\mathbf{F}=G M m \frac{\mathbf{r}}{|\mathbf{r}|^{3}}=G M m \frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}$$ where \(G\) is the gravitational constant. a. Verify that this force field is conservative on any region excluding the origin. b. Find a potential function \(\varphi\) for this force field such that \(\mathbf{F}=-\nabla \varphi\). c. Suppose the object with mass \(m\) is moved from a point \(A\) to a point \(B\), where \(A\) is a distance \(r_{1}\) from \(M,\) and \(B\) is a distance \(r_{2}\) from \(M .\) Show that the work done in moving the object is \(G M m\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)\). d. Does the work depend on the path between \(A\) and \(B ?\) Explain.

Short answer

\)$$ Answer: No, the work done in moving an object with mass m from point A to point B is not dependent on the path taken in the given force field since it is a conservative force field.

Step-by-step solution

Verifying the given force field is conservative

To verify if the given force field is conservative, we need to check if the curl of the force field is zero. The curl of a vector field is given by: $$\text{curl} \, \mathbf{F} = \nabla \times \mathbf{F}$$ So, let's compute the curl of the given force field: $$\text{curl} \, \mathbf{F} = \nabla \times \left(G M m \frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} \right)$$ After calculating the curl, we find: $$\text{curl} \, \mathbf{F} = \langle 0, 0, 0 \rangle$$ Since the curl of the force field is zero, the force field is conservative.

Finding the potential function

Now, we need to find a potential function φ for the conservative force field. To do this, we will use the relation: $$\mathbf{F}=-\nabla \varphi$$ Integrating the partial derivatives of φ with respect to x, y, and z, we obtain: $$\varphi = -G M m \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}} + C$$ Where C is a constant of integration.

Calculating the work done

To find the work done in moving the object with mass m from point A to point B, we will use the potential function we found in the previous step. The work done is given by: $$W = \varphi(B) - \varphi(A)$$ Substituting the potential function at points A and B, where A is a distance \(r_1\) from M, and B is a distance \(r_2\) from M, we have: $$W = \left(-G M m \frac{1}{r_2} + C\right) - \left(-G M m \frac{1}{r_1} + C\right)$$ Simplifying the expression, we obtain: $$W = G M m\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$$

Determining if the work done depends on the path

Since we have already found that the given force field is conservative, and the work done is evaluated as a difference in the potential function, we can conclude that the work done does not depend on the path between points A and B. This is a property of conservative force fields, as they can be represented as the gradient of a scalar potential function.

Key concepts

Conservative Force Fields

In physics, understanding the behavior of forces is crucial for predicting the movement of objects and energy interactions. One such type of force is known as a 'conservative force.' A conservative force field is one where the work done by the force on an object moving from one point to another is the same, regardless of the path taken between the points.

Conservative forces have a critical property: they conserve mechanical energy. That means that within a conservative force field, the sum of kinetic and potential energy remains constant when no other external work is done or non-conservative forces are present.

Gravity is a perfect example of a conservative force — the work done by gravitational forces will only depend on the initial and final positions, not on the path taken. In mathematical terms, a force field is conservative if its curl is equal to zero. This can be represented as: \[\text{curl} \, \mathbf{F} = abla \times \mathbf{F} = \mathbf{0}\]. When analyzing the gravitational force in the example given, we verified its conservative nature by calculating the curl and confirming it equated to zero vector.

Potential Function

Associated with every conservative force field is a corresponding potential function. This function, often denoted by \(\varphi\), represents the potential energy per unit charge or mass at each point in the force field. It provides a scalar value that, when differentiated, gives us the force field itself. For a gravitational force field, the potential function is crucial in calculating the work done by the gravitational force.

From the perspective of calculus, the force field can be expressed as the negative gradient of the potential function: \[\mathbf{F}=-abla \varphi\]. The negative sign indicates that the force is acting in the direction of decreasing potential. In the given exercise, by integrating the force field, we derived the potential function for the gravitational force field: \[\varphi = -G M m \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}} + C\], where \(C\) is a constant that represents the arbitrary zero level of potential energy.

Work Done in Physics

Work done is a foundational concept in physics, describing the process of energy transfer when a force is applied to an object causing displacement. For a conservative force field, work done can be elegantly calculated using the potential function.

The work done, denoted by \(W\), is the difference in the potential energy of the object at two different points. It is found by taking the difference in the potential function's values at the starting and ending points of the motion: \[W = \varphi(B) - \varphi(A)\]. In the context of gravitational forces, this means the work done in moving an object from point \(A\) with a distance \(r_1\) from a mass \(M\), to point \(B\) with a distance \(r_2\), is independent of the actual path taken, adhering to the basic properties of conservative forces.

Using the established potential function from the earlier section, the work done was calculated as \(W = G M m(\frac{1}{r_{2}}-\frac{1}{r_{1}})\), reiterating that in a conservative field like gravity, the only factors that matter are the initial and final positions - not the trajectory between them.