Chapter 8, Problem 8.13P

Consider a pulsar, a collapsed star of extremely high density, with a mass M equal to that of the Sun (1.98 × 1030 kg), a radius R of only 17.4 km, and a rotational period T of 0.0448 s. By what percentage does the free-fall acceleration g differ from the gravitational acceleration ag at the equator of this spherical star?

Using Figure 13.9, carefully sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass, m. Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal acceleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle ε , defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. tan(λ + ε) = (g / (g − ω2 RE)) . tanλ , where ω is the angular velocity of Earth.

If a particle be describing an ellipse about a centre of force in the centre, show that the sum of the reciprocates of its angular velocities about foci is constant.