4.6 Problem Set

Cavendish claimed to be “weighing the earth” by his experiment. Using Newton’s second law in combination with his law of universal gravitation, and using the facts that the acceleration due to gravity at the surface of the earth is g = 9.81 m/s2, and the radius of the earth is 6368 km, compute the mass of the earth. (In solving this problem, assume the mass is all at a point in the center of the body. It is a remarkable truth, as proved by Gauss, that as long as the bodies possess spherical symmetry, this gives the correct answer.)

Determine the repulsive force between two electrons placed at a distance of 1 mm from each other.

Determine the attractive force due to gravitation between the above two electrons (mass of electron \(- 9.107 \times 10^{-31}\)kg). Hence demonstrate that for electron the electrical force is about 1042 times as strong as the gravitational force.

Compute the total magnetic field at the point in Fig.4.4 by summing (integrating) the contribution from all elements \(dl\), in a very long wire carrying current \(i\). (Assume the point is a perpendicular distance \(R\) from the wire.)

Determine the potential energy stored in a spring stretched from equilibrium a distance \(x\). (Recall, Hooke’s Law is \(F = kx\).) Hooke’s Law is, as you know, the same force law as that between two molecules at distances very near their equilibrium distance. This potential energy is a way of representing the vibrational energy of a diatomic molecule, which in the course of vibrating is rapidly exchanging potential energy (maximum at the maximum distance from equilibrium) for kinetic energy (maximum at the equilibrium distance).

Show that \(\frac{GM}{R^2}=g\), the acceleration due to gravity at the surface of the earth. Hence show that the escape velocity is approximately seven miles per second, knowing that \(g\) = 9.81 m/s2 and \(R\) = 6368 km

A glaucous-winged gull weighing 2 kg carries a cockle weighing 0.25 kg to a height of 10 m where at a horizontal flight speed of 3 m/sec it releases the cockle whose shell will shatter on the rocks below, as soon thereafter as possible to be devoured by the gull. What is the minimum energy that the gull could have expended in rising to 10 m and achieving its level flight?

At what speed will the cockle hit the ground?

The lowest energy state for the electron in the hydrogen atom has been measured to be -217.3 \(\times\) 10-20 joules. What is the radius of the hydrogen atom under this condition? (HINT: The centrifugal force of a mass moving in a circular orbit is \(\frac{mv^2}{r}\). When this is equated to the electrostatic force holding the electron in orbit, a simple expression for the kinetic energy of the electron is obtained. Add this to the potential energy to get the total energy, which is -217.3 \(\times\) 10-20 joules. Solve for \(r\).)