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Ch. 1 Problems

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Basic

Medium

Basic

B-1. Estimating the Mass of Neptune

Consider the problem of estimating the mass of an unknown planet (Neptune) from the deviation in the orbit of Uranus. Let the orbital radius of Uranus be \(r_U\), the orbital radius of Neptune be \(r_N\), and the observed deviation in acceleration be \(\delta a\). Express the mass of Neptune \(M_N\) in terms of \(\delta a\), \(r_U\), \(r_N\), and \(G\).

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B-2. Why T-V and Not T+V

Consider the motion of a ball of mass \(m\) thrown vertically upward. Let \(y(t)\) be the height, with \(T = \frac{1}{2}m\dot{y}^2\) and \(V = mgy\).

(a) Taking the Lagrangian as \(L = T - V\), apply the Euler-Lagrange equation \(\frac{d}{dt}\frac{\partial L}{\partial \dot{y}} - \frac{\partial L}{\partial y} = 0\) and verify that Newton's equation of motion \(m\ddot{y} = -mg\) is obtained.

(b) Instead, take \(L' = T + V = \frac{1}{2}m\dot{y}^2 + mgy\) and apply the same Euler-Lagrange equation. Verify that the unphysical result \(m\ddot{y} = +mg\) (gravity pointing upward!) is obtained.

Key Point

Using \(T + V\) leads to the result that "the ball accelerates upward," which contradicts experiment. The reason it must be \(T - V\) is that "otherwise it does not agree with experiment"—that is, the form of the Lagrangian is a hypothesis that is verified by experiment.

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Medium

M-1. Derivation of Kepler's Third Law

Using the gravitational force \(F = GMm/r^2\) and the circular motion condition \(F = mv^2/r\), derive Kepler's Third Law \(T^2 \propto r^3\) for the case of a circular orbit.

Hint

The period of circular motion is \(T = 2\pi r / v\).

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M-2. Calculation of the Gravitational Potential

A point particle of mass \(M\) is placed at the origin. Solve the Poisson equation \(\nabla^2 \Phi = 4\pi G \rho\) under the assumption of spherical symmetry and derive \(\Phi = -GM/r\).

Hint

The Laplacian in spherical coordinates is \(\nabla^2 \Phi = \frac{1}{r^2}\frac{d}{dr}\left(r^2 \frac{d\Phi}{dr}\right)\) (with \(\rho = 0\) in the region \(r \neq 0\)).

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