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Prologue Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Calculating the Gravitational Force Between the Sun and Earth

Using Newton's law of universal gravitation \(F = Gm_1 m_2/r^2\), calculate the magnitude of the gravitational force between the Sun (mass \(M_\odot \approx 2.0 \times 10^{30}\) kg) and the Earth (mass \(M_\oplus \approx 6.0 \times 10^{24}\) kg). Take the Sun–Earth distance to be \(r \approx 1.5 \times 10^{11}\) m.

Hint

Simply substitute the values into \(F = G M_\odot M_\oplus / r^2\). Since there are many digits involved, it is easier to first combine the powers of ten separately.

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B-2. Ratio of Gravitational to Coulomb Force Between Protons

Calculate the ratio of the gravitational force to the Coulomb force between two protons, \(F_{\text{grav}}/F_{\text{em}}\), by substituting specific numerical values. Use \(G \approx 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2\), \(m_p \approx 1.67 \times 10^{-27}\) kg, \(e \approx 1.60 \times 10^{-19}\) C, and \(1/(4\pi\varepsilon_0) \approx 9.0 \times 10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2\).

Hint

\(F_{\text{grav}} = G m_p^2 / r^2\), \(F_{\text{em}} = e^2/(4\pi\varepsilon_0 r^2)\). When taking the ratio, \(r^2\) cancels out.

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B-3. Gradient of a 2D Potential

For the two-dimensional potential \(\Phi(x, y) = x^2 + 4y^2\), answer the following questions.

(a) Find the gradient \(\nabla\Phi = (\partial\Phi/\partial x,\;\partial\Phi/\partial y)\).

(b) Find the direction and magnitude of the gradient vector at the point \((1, 1)\).

(c) In what direction does the force \(\boldsymbol{F} = -m\nabla\Phi\) acting on an object at this point point? Explain in terms of its relationship to the equipotential lines (curves where \(\Phi = \text{const}\)).

Hint

(a) Simply take the partial derivative with respect to each variable. (b) The magnitude of the vector is \(|\nabla\Phi| = \sqrt{(\partial\Phi/\partial x)^2 + (\partial\Phi/\partial y)^2}\). (c) In what direction does the gradient vector point relative to the equipotential lines?

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B-4. Relativistic Parameter for Various Celestial Bodies

Calculate the criterion \(\Phi_{\text{rel}} = GM/(Rc^2)\) for the following celestial bodies.

(a) The Sun (\(M_\odot \approx 2.0 \times 10^{30}\) kg, \(R_\odot \approx 7.0 \times 10^{8}\) m)

(b) A neutron star (\(M \approx 1.4\,M_\odot\), \(R \approx 10\) km)

Express each result to 1 significant figure and discuss how reliable Newton's model is in each case.

Hint

Use \(c \approx 3.0 \times 10^8\) m/s, and first calculate the numerator \(GM\) before dividing by \(Rc^2\).

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B-5. Derivation of the Schwarzschild Radius

Using the escape velocity \(v_{\text{esc}} = \sqrt{2GM/R}\), express the radius \(R_s\) at which \(v_{\text{esc}} = c\) in terms of \(M\), \(G\), and \(c\). This \(R_s\) is called the Schwarzschild radius.

Hint

Substitute \(v_{\text{esc}} = c\) and solve for \(R\).

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B-6. Criterion at the Schwarzschild Radius

Using the Schwarzschild radius \(R_s\) obtained in Problem B-5. Derivation of the Schwarzschild Radius, calculate the criterion \(GM/(Rc^2)\) at \(R = R_s\) and verify the value corresponding to a black hole.

Hint

Substitute the expression for \(R_s\) into \(GM/(Rc^2)\).

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B-7. Equivalence Condition of Inertial and Gravitational Mass

Distinguishing between inertial mass \(m_i\) and gravitational mass \(m_g\), we write the equation of motion in a uniform gravitational field \(\mathbf{g}\) as

\[ m_i\,\ddot{\boldsymbol{x}} = m_g\,\mathbf{g} \]

Find the condition, expressed in terms of the ratio of \(m_i\) to \(m_g\), for object A (\(m_i^{(A)},\; m_g^{(A)}\)) and object B (\(m_i^{(B)},\; m_g^{(B)}\)) to fall with the same acceleration.

Hint

Write down the acceleration of each object \(\ddot{\boldsymbol{x}} = (m_g/m_i)\,\mathbf{g}\), and consider the condition for both to be equal.

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B-8. Basic Calculations of Partial Derivatives

Practice calculating partial derivatives. For the following functions, find the specified partial derivatives.

(a) For \(f(x, y) = 3x^2 y + 2y^3\), find \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\).

(b) For \(g(x, y, z) = x^2 y z^3\), find \(\dfrac{\partial g}{\partial x}\), \(\dfrac{\partial g}{\partial y}\), and \(\dfrac{\partial g}{\partial z}\).

(c) For \(h(r, \theta) = r^2 \cos\theta\), find \(\dfrac{\partial h}{\partial r}\) and \(\dfrac{\partial h}{\partial \theta}\).

Hint

In partial differentiation, all variables other than the one being differentiated with respect to are treated as constants. For example, in \(\partial(3x^2 y)/\partial x\), we treat \(y\) as a constant and differentiate with respect to \(x\).

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B-9. Gradient Vectors and Isotherms

The relationship between partial derivatives and the gradient. The function \(T(x, y) = 100 - x^2 - 4y^2\) of two variables represents the temperature distribution on a certain plane.

(a) Find \(\dfrac{\partial T}{\partial x}\) and \(\dfrac{\partial T}{\partial y}\).

(b) Calculate the values of \(\partial T/\partial x\) and \(\partial T/\partial y\) at the point \((1, 2)\), and explain in words their physical meaning ("how does the temperature change when you move slightly in the \(x\) direction?").

(c) Calculate the gradient vector \(\nabla T = (\partial T/\partial x,\;\partial T/\partial y)\) at the point \((1, 2)\), and state the direction this vector points relative to the isotherms (curves where \(T = \text{const.}\)).

Hint

(b) Simply substitute the numerical values. Pay attention to the sign—if negative, it means "the temperature decreases as you move in that direction." (c) The gradient vector is always perpendicular to the isotherms and points in the direction of the steepest temperature increase.

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Medium

M-1. Quantitative Evaluation of the Four Properties of Gravity

Based on the four fundamental properties of gravity (universality, impossibility of shielding, long-range nature, and extreme weakness), answer the following questions.

(a) Quantitatively explain why gravity is negligible at the atomic/molecular scale (\(\sim 10^{-10}\) m), using the ratio of gravitational to electromagnetic force strengths.

(b) Explain why gravity nevertheless becomes dominant at the scale of galaxy clusters (\(\sim 10^{23}\) m), clearly stating which of the four properties are essential to this explanation.

Hint

For (a), make use of the ratio \(F_{\text{grav}}/F_{\text{em}} \sim 10^{-36}\) given in the text. For (b), contrast the reason why electromagnetic forces are neutralized at large scales with the reason why gravity is not neutralized.

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M-2. Stages of the Transition from Newton to GR

For the celestial bodies listed in the table in the main text (Earth, Sun, white dwarf, neutron star, black hole), discuss how deviations from Newton's model become manifest as the value of \(GM/(Rc^2)\) increases, correlating them with specific observational phenomena (perihelion precession of Mercury, GPS time corrections, formation of the event horizon, etc.) (approximately 5–8 sentences).

Hint

Organize the stages according to the magnitude of \(GM/(Rc^2)\): "Newton is sufficient" → "small corrections are needed" → "Newton completely breaks down."

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M-3. The Relationship Between Geodesics and the Equivalence Principle

In Einstein's general relativity, "gravity is not a force but the curvature of spacetime." Explain in 3–5 sentences why, under this interpretation, the equivalence of inertial mass and gravitational mass (\(m_i = m_g\)) is no longer an "unexplained coincidence," from the perspective that "all objects move along geodesics in curved spacetime."

Hint

Note that geodesics are determined solely by the geometry of spacetime and do not depend on the mass or composition of the moving object.

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Advanced

A-1. Gravitational Time Dilation of GPS Satellites

A GPS satellite orbits in a circular orbit at an altitude of approximately \(h \approx 2.0 \times 10^{4}\) km. Take the mass of the Earth as \(M_\oplus \approx 6.0 \times 10^{24}\) kg and the radius of the Earth as \(R_\oplus \approx 6.4 \times 10^{3}\) km.

(a) Find the difference in gravitational potential between the Earth's surface and the satellite orbit: \(\Delta\Phi = \Phi(R_\oplus + h) - \Phi(R_\oplus)\).

(b) According to general relativity, between locations with a gravitational potential difference \(\Delta\Phi\), there is a relative shift in the rate at which time passes. This ratio is approximately given by

\[ \frac{\Delta\tau}{\tau} \approx \frac{\Delta\Phi}{c^2} \]

(if \(\Delta\tau/\tau > 0\), the satellite clock runs faster). Using this expression, estimate how many microseconds (\(\mu\)s) per day the satellite clock runs faster than a clock on the ground.

(c) Using the result from (b), estimate the magnitude of the position error that would accumulate per day if this time offset were not corrected.

Hint

(a) Use \(\Phi(r) = -GM_\oplus/r\) and take the difference between the two points. (b) Multiply by 1 day \(= 86400\) seconds. (c) The product of the speed of light \(c\) and the time offset gives an estimate of the distance error. Note that in this problem, the special relativistic effect (time dilation due to the satellite's motion) is neglected.

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