Skip to content

Ch. 1 Problems

← Back to chapter　|　View solutions

Table of Contents

Basic

Medium

Advanced

Basic

B-1. Calculating Gravitational Acceleration at Earth's Surface

Using Earth's mass \(M_\oplus \approx 5.97 \times 10^{24}\ \mathrm{kg}\) and radius \(R_\oplus \approx 6.37 \times 10^6\ \mathrm{m}\), calculate the magnitude of the gravitational field at Earth's surface \(|\mathbf{g}| = GM_\oplus/R_\oplus^2\), and confirm that it agrees with the gravitational acceleration \(g \approx 9.8\ \mathrm{m/s^2}\) learned in high school physics.

Hint

Substitute \(G \approx 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}\) and carry out the calculation.

→ View solution

B-2. Differentiation of the \(x\) Component of the Gravitational Potential

When a mass \(M\) is located at the origin, the gravitational potential is \(\Phi = -GM/r\). Using Cartesian coordinates \((x, y, z)\) with \(r = \sqrt{x^2 + y^2 + z^2}\), calculate \(\partial \Phi / \partial x\) and find the \(x\) component of the gravitational field \(g_x = -\partial \Phi / \partial x\).

Hint

Use the chain rule to differentiate, applying \(\partial r / \partial x = x/r\).

→ View solution

B-3. Vector Representation of the Gravitational Field

Extend the result of Problem B-2. Differentiation of the \(x\) Component of the Gravitational Potential to the \(y\)- and \(z\)-components, and show in vector form that \(\mathbf{g} = -\nabla\Phi\) is consistent with Eq. (1.3), \(\mathbf{g} = -GM\,\hat{\mathbf{r}}/r^2\).

Hint

Use the fact that \(\hat{\mathbf{r}} = (x/r,\; y/r,\; z/r)\).

→ View solution

B-4. Superposition of Two Point Masses

Using the superposition principle, write down the composite gravitational potential produced by two point masses \(M_1\) (placed at the origin) and \(M_2\) (placed at position \(\mathbf{r}_0\)). Furthermore, find the gravitational field \(\mathbf{g}(\mathbf{r})\) at position \(\mathbf{r}\).

Hint

Since the Poisson equation is linear, the total potential is the sum of the potentials produced by each mass.

→ View solution

B-5. Calculating \(\nabla^2(r^n)\)

The radial part of the Laplacian in spherical coordinates \((r, \theta, \varphi)\) is

\[ \nabla^2 f(r) = \frac{1}{r^2}\frac{d}{dr}\!\left(r^2 \frac{df}{dr}\right) \]

For \(f(r) = r^n\) (where \(n\) is an integer), calculate \(\nabla^2(r^n)\) and organize the result in terms of \(n\).

Hint

Substitute \(df/dr = n\,r^{n-1}\), then differentiate \(r^2 \cdot n\,r^{n-1}\) once more with respect to \(r\).

→ View solution

B-6. Laplace Equation Outside a Point Mass

Using the result from Problem B-5. Calculating \(\nabla^2(r^n)\), verify that for \(\Phi = -GM/r = -GM\,r^{-1}\), we have \(\nabla^2 \Phi = 0\) at \(r \neq 0\). Explain why this result does not contradict the Poisson equation \(\nabla^2 \Phi = 4\pi G\rho\).

Hint

Note that for \(r \neq 0\), the mass density of a point particle \(\rho = M\,\delta^3(\mathbf{r})\) is zero.

→ View solution

B-7. Potential Constant Inside a Uniform Density Sphere

Assume that inside a sphere (radius \(R\)) of uniform density \(\rho_0\) (constant), the potential takes the form \(\Phi(r) = Ar^2 + B\) (\(A, B\) are constants). By substituting into the Poisson equation \(\nabla^2 \Phi = 4\pi G\rho_0\), express the constant \(A\) in terms of \(G\) and \(\rho_0\).

Hint

Use the result from Problem B-5. Calculating \(\nabla^2(r^n)\) for \(\nabla^2(r^2)\) when \(n = 2\).

→ View solution

B-8. Divergence of the Gravitational Field and Poisson's Equation

Express the divergence \(\nabla \cdot \mathbf{g}\) of the gravitational field \(\mathbf{g} = -\nabla\Phi\) in terms of \(\rho\) using Poisson's equation.

Hint

We can write \(\nabla \cdot \mathbf{g} = \nabla \cdot (-\nabla\Phi) = -\nabla^2\Phi\).

→ View solution

B-9. Contradiction Between Instantaneous Propagation and Special Relativity

In Newton's gravitational model, if the Sun were to suddenly disappear, the Earth would immediately begin moving in a straight line at that instant. Calculate the time it takes for light to travel from the Sun to the Earth (approximately 8 minutes), and explain the contradiction between Newton's model and special relativity using concrete numerical values.

Hint

Divide the Sun–Earth distance \(\approx 1.5 \times 10^{11}\) m by the speed of light \(c \approx 3.0 \times 10^8\) m/s.

→ View solution

B-10. Estimating Relativistic Effects at the Solar Surface

Using the Sun's mass \(M_\odot \approx 1.99 \times 10^{30}\ \mathrm{kg}\) and radius \(R_\odot \approx 6.96 \times 10^8\ \mathrm{m}\), calculate \(GM_\odot/(R_\odot c^2)\). From this value, estimate the magnitude of relativistic effects near the solar surface.

Hint

Using \(c \approx 3.0 \times 10^8\) m/s, first calculate the numerator \(GM_\odot\), then divide by \(R_\odot c^2\).

→ View solution

B-11. Estimating the Relativistic Parameter of a Neutron Star

For the dimensionless quantity \(GM/(Rc^2)\), use the typical mass \(M \approx 1.4\,M_\odot\) and radius \(R \approx 10\ \mathrm{km}\) of a neutron star. Express this quantity in terms of \(G\), \(M_\odot\), \(R\), and \(c\), and estimate the order of magnitude (i.e., the power of \(10\)).

Hint

Using the relation \(GM_\odot/c^2 \approx 1.48\ \mathrm{km}\) makes the calculation straightforward.

→ View solution

Medium

M-1. Derivation of Poisson's Equation from Gauss's Law

Gauss's law for the gravitational field can be written for the total mass \(M_{\mathrm{enc}}\) contained within a region \(V\) enclosed by a closed surface \(S\) as

\[ \oint_S \mathbf{g} \cdot d\mathbf{A} = -4\pi G\,M_{\mathrm{enc}} \]

Using \(\mathbf{g} = -\nabla\Phi\) and the divergence theorem, derive Poisson's equation \(\nabla^2\Phi = 4\pi G\rho\) from this integral form.

Hint

Apply the divergence theorem \(\oint_S \mathbf{g} \cdot d\mathbf{A} = \int_V \nabla \cdot \mathbf{g}\; dV\) and use \(M_{\mathrm{enc}} = \int_V \rho\; dV\). Since the equality must hold for an arbitrary volume \(V\), you can obtain the differential form.

→ View solution

M-2. Complete Potential Solution for a Uniform Density Sphere

For a sphere of radius \(R\), uniform density \(\rho_0\), and total mass \(M = \frac{4}{3}\pi R^3 \rho_0\), do the following.

(a) Show that outside the sphere (\(r > R\)), \(\Phi_{\mathrm{out}}(r) = -GM/r\) by solving the spherically symmetric Poisson equation (with \(\rho = 0\)).

(b) Solve the Poisson equation inside the sphere (\(r < R\)) to find \(\Phi_{\mathrm{in}}(r)\). Use the boundary conditions that the potential and its derivative \(d\Phi/dr\) are continuous at \(r = R\).

(c) Find the value of \(\Phi\) at \(r = 0\) and compare it with the value at the surface \(\Phi(R)\).

Hint

For the interior, assume the form \(\Phi = Ar^2 + B\) (using the result from Problem B-7. Potential Constant Inside a Uniform Density Sphere), and determine \(A\) and \(B\) from the two matching conditions at \(r = R\).

→ View solution

M-3. Scale Estimation of Mercury's Perihelion Precession

Given Mercury's orbital semi-major axis \(a \approx 5.79 \times 10^{10}\ \mathrm{m}\), calculate the dimensionless quantity \(GM_\odot/(ac^2)\). Using dimensional analysis, discuss how this value corresponds in order of magnitude to the perihelion precession "deviation from the Newtonian model" of 43 arcseconds per century (\(\approx 2.1 \times 10^{-7}\ \mathrm{rad}\)).

Hint

Mercury's orbital period is approximately 88 days, so estimate the number of orbits in 100 years and compare the precession angle per orbit with \(GM_\odot/(ac^2)\).

→ View solution

M-4. Comparison of the Wave Equation and Poisson's Equation

In the electromagnetic wave equation

\[ \left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\varphi = -\frac{\rho_e}{\varepsilon_0} \]

show what form this equation reduces to when the source \(\rho_e\) is time-independent (electrostatic field). Furthermore, organize the structural similarities and differences with Newton's Poisson equation.

Hint

Setting \(\partial/\partial t = 0\) eliminates the time derivative term. Compare the remaining equation with the electrostatic Poisson equation.

→ View solution

Advanced

A-1. An Attempt at a Scalar Theory of Gravity

By adding a time derivative to the Poisson equation \(\nabla^2\Phi = 4\pi G\rho\), we formally obtain

\[ \left(\nabla^2 - \frac{1}{c_g^2}\frac{\partial^2}{\partial t^2}\right)\Phi = 4\pi G\rho \tag{$\ast$} \]

which gives a "gravitational wave equation" in which changes in gravity propagate at speed \(c_g\).

(a) Setting \(c_g = c\) (the speed of light), find the dispersion relation \(\omega(\mathbf{k})\) for the plane wave solution \(\Phi = \Phi_0\,e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}\) of this equation in the source-free case (\(\rho = 0\)).

(b) Although this modification resolves the "instantaneous propagation" problem of the Newtonian model, this scalar gravity theory actually has another serious problem. By contrasting with electromagnetism, where the field is described by a vector potential \(A^\mu\), discuss physically the limitations of describing gravity with only a scalar potential \(\Phi\). (Hint: Focus on the physical quantity that serves as the source. Recall that in special relativity, energy and momentum are unified.)

(c) Show that taking the limit \(c_g \to \infty\) in equation \((\ast)\) reduces it to the Poisson equation, and explain how this is consistent with the statement that "Newtonian gravity is an approximation in the \(c \to \infty\) limit."

Hint

(a) Substitute the plane wave and find the relationship between \(\omega\) and \(|\mathbf{k}|\). (b) Consider that in special relativity, the energy-momentum tensor \(T^{\mu\nu}\) should serve as the source of gravity. Discuss why the scalar \(\rho\) alone is insufficient. (c) Simply take \(1/c_g^2 \to 0\).

→ View solution

A-2. Shell Theorem and Tidal Force

Show that the gravitational potential \(\Phi\) is constant inside the hollow cavity (\(r < R_1\)) of a uniform-density spherical shell (inner radius \(R_1\), outer radius \(R_2\)) using the Poisson equation and appropriate boundary conditions (Newton's shell theorem).

Furthermore, using this result, discuss the following:

(a) If a mass \(m\) is placed at a position \(\mathbf{r}_0\) slightly displaced from the center of the shell, is the gravitational force on the object zero? Explain why.

(b) If the shell is not perfectly spherically symmetric but slightly deformed into an ellipsoid, the potential inside the cavity is no longer constant. Qualitatively discuss the nature of the gravitational field that arises inside the cavity, and explain how this relates to the concept of tidal force. (Hint: In general relativity, tidal forces are described as spacetime curvature. What is the corresponding quantity in Newtonian gravity?)

Hint

Inside the shell, \(\rho = 0\), so \(\nabla^2\Phi = 0\). The only spherically symmetric solution that is regular at \(r = 0\) is a constant. For (b), note that the second derivatives of \(\Phi\) (the spatial variation of the gravitational field) correspond to tidal forces.

→ View solution