Question

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

Question: Verify if the gravitational force field is conservative. If so, find the potential function φ for this force field and calculate the work done when moving a mass m from point A to point B. Check if the work depends on the path between A and B. Answer: The gravitational force field is conservative, the potential function is φ(x,y,z) = - GMm(1/√(x^2+y^2+z^2)) + C, and the work done to move the mass m from point A to point B is W = GMm(1/r₂ - 1/r₁). This work does not depend on the path between A and B, but only on the initial and final positions of the mass m.

Step-by-step solution

a. Verifying the force field is conservative

To determine if the force field is conservative, we need to compute its curl and check if the curl is zero. The curl of a vector field \(\mathbf{F} = \langle P, Q, R \rangle\) is computed by: $$ \nabla \times \mathbf{F} = \langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\rangle $$ Let's compute the curl of the given force field where \(P = GMm\frac{x}{(x^2+y^2+z^2)^{3/2}}\), \(Q = GMm\frac{y}{(x^2+y^2+z^2)^{3/2}}\), and \(R = GMm\frac{z}{(x^2+y^2+z^2)^{3/2}}\).

1. Compute Partial Derivatives for the Curl

Compute all the partial derivatives needed for the curl: $$ \frac{\partial P}{\partial y} \quad\frac{\partial P}{\partial z} \quad\frac{\partial Q}{\partial x} \quad\frac{\partial Q}{\partial z} \quad\frac{\partial R}{\partial x} \quad\frac{\partial R}{\partial y} $$ In order to ease calculations, we first factor out \(GMm\), having all expressions multiplied by this.

2. Calculate the Curl of the Force Field

Now, we can plug in all our partial derivatives to complete the curl calculation: $$ \nabla \times \mathbf{F} = \langle 0, 0, 0\rangle $$ Since the curl of the gravitational force field is zero, we can conclude that the force field is conservative on any region excluding the origin.

b. Find a potential function

We are given that \(\mathbf{F} = -\nabla \varphi\), so we can determine the potential function. To do this, we integrate each component of the force field with respect to each coordinate. $$ \varphi = - \int P\,dx - \int Q\,dy - \int R\,dz $$ By solving the three integration parts, we get: $$ \varphi(x,y,z) = - GMm \frac{1}{\sqrt{x^2+y^2+z^2}} + C $$ Where C is integration constant. So, we found our potential function.

c. Calculate work done to move the object from A to B

The work done by a conservative force when moving an object between two points A and B is equal to the negative of the change in the potential energy between those points: $$ W = -\Delta \varphi = -(\varphi(B)-\varphi(A)) $$ Using the potential function we found, we can compute the work done by the gravitational force when moving object m from point A(\(r_1\)) to point B(\(r_2\)): $$ W = -(- GMm \left( \frac{1}{r_2} - \frac{1}{r_1} \right)) = GMm\left(\frac{1}{r_{2}} - \frac{1}{r_{1}}\right) $$ Hence, the work done is \(GMm\left(\frac{1}{r_{2}} - \frac{1}{r_{1}}\right)\).

d. Path dependence

Since we found that the force field is conservative and calculated the work from the potential function, we can know that the work done by a conservative force does not depend on the path taken between points A and B. The work only depends on the initial and final positions of the object. In this case, the work done to move the object with mass m from point A to point B depends only on the distances \(r_1\) and \(r_2\) and not on the actual path taken.

Key concepts

Conservative Force Fields

When studying forces in physics, understanding the concept of a 'conservative force field' is fundamental. A conservative force field is characterized by its ability to conserve mechanical energy within a system. This means that as an object moves through a conservative force field, such as the gravitational field described in the exercise, the sum of its kinetic and potential energies remains constant.

To verify that a force field is conservative, one must check whether the curl of the force field is zero in a simply-connected region, excluding any singularities like the origin in our gravitational force example. In the context of the exercise, after calculating the curl of the gravitational force and finding it to be zero, this affirmation of the force being a conservative field is validated. This confirmation simplifies many physical predictions, as certain properties are known to follow, including path independence of work done by the force, which is a direct consequence of the field's conservativeness.

Potential Function

A 'potential function' is a helpful tool in physics to address the effect of conservative force fields without directly dealing with the forces themselves. This potential function, represented by \( \varphi \), relates to the force field by a simple negative gradient relationship, wherein \( \mathbf{F} = -abla \varphi \). The gradient operation involves taking the spatial derivatives of the potential function, and its negative provides the force at any point in space.

In our gravitational problem, finding the potential function involves integrating the components of the force field, which yields a scalar field that depends on position. By doing so, we get an expression for the gravitational potential energy relative to the mass m, which, minus the constant of integration, defines the potential function. This potential simplifies many analyses, like calculating the work done on an object moving between two points, which is precisely what the potential function was used for in the exercise.

Work Done by a Force

In physics, 'work done by a force' is a measure of energy transfer when a force acts over a distance. Crucially, if the force is conservative, like the gravitational force we've been examining, the work done only depends on the start and endpoints—not the path taken. This path independence is because, in conservative fields, any energy expended against the force is stored as potential energy, which can be fully recovered.

In the context of the exercise, the work done in moving an object with mass m from point A to point B in a gravitational field is simply the change in gravitational potential energy between the two points. This outcome, represented by the equation \( W = GMm\left(\frac{1}{r_{2}} - \frac{1}{r_{1}}\right) \), illustrates that work is independent of the trajectory taken from A to B. This principle has profound implications, ensuring that, for instance, satellite transfers between orbits or the motion of planets around the sun involves predictable energy changes.