Question 1

At which point in the elliptical orbit (A, B, C, D, or E) is the speed of the object at its maximum, and why?

Accepted Answer

Point A, because it is closest to the Earth and gravitational potential energy is at a minimum, so kinetic energy is at a maximum.

Question 2

At point B in the orbit, the gravitational force vector can be split into two components relative to the velocity vector. What is the effect of the component of force parallel to the velocity?

Accepted Answer

It decreases the speed of the object.

Question 3

Which of the following best describes the angular momentum of the object at points A and C?

Accepted Answer

$$L_A = L_C$$, because angular momentum is conserved in the absence of external torques.

Question 4

Given $$r_A ($$distance to Earth at A), $$v_A ($$speed at A), and $$m ($$mass of object), what is the angular momentum at point A?

Accepted Answer

$$L_A = m v_A r_A$$

Question 5

If a rocket thruster at point E exerts a brief force in the direction of motion, what happens to the object’s angular momentum about the center of the Earth?

Accepted Answer

It increases.

Question 6

To achieve escape speed with the minimum amount of chemical energy at point E, in which direction should the rocket thruster push?

Accepted Answer

In the direction of the velocity vector (tangential to the orbit).

Question 7

The escape speed $$v_{esc}$$ from a distance $$r$$ from the center of the Earth is given by:

Accepted Answer

$$v_{esc} = \sqrt{2GM/r}$$

Question 8

Suppose the rocket at point E increases the speed from $$v_E to$$ $$\alpha v_E$$ ($$\alpha \u003e 1). $$What is the value of $$\alpha$$ required for escape, in terms of $$M$$, $$r_E$$, and $$v_E$$?

Accepted Answer

$$\alpha = \sqrt{\frac{2GM}{r_E v_E^2$}}$$

Question 9

Why does a thruster blast in a direction other than the velocity vector require more chemical energy to achieve escape speed?

Accepted Answer

Because only the component of the force in the direction of motion increases the kinetic energy most efficiently.

Question 10

Newton’s law of universal gravitation states that the force between two point masses is:

Accepted Answer

$$F = G \frac{m_1$ m_2$}{r^2$}$$

Question 11

What is the gravitational force on a point mass located anywhere inside a spherically symmetric shell of mass?

Accepted Answer

Zero, because the forces from all parts of the shell cancel out.

Question 12

For a point mass $$m$$ outside a uniform spherical shell of mass $$M$$ and radius $$R$$, the gravitational force is:

Accepted Answer

$$F = G \frac{mM}{r^2$}$$, as if all mass were concentrated at the center.

Question 13

The gravitational potential energy $$U_g$$ for a point mass $$m at $$distance $$r$$ from a mass $$M is$$:

Accepted Answer

$$U_g = -G \frac{mM}{r}$$

Question 14

A ball is dropped into a tunnel through the Earth (assume uniform density, no air resistance). What type of motion does it undergo?

Accepted Answer

Simple harmonic motion, like a mass on a spring.

Question 15

What is the effective 'spring constant' $$k$$ for the gravitational force inside the Earth (uniform density, mass $$M$$, radius $$R$$)?

Accepted Answer

$$k = \frac{G M m}{R^3$}$$

Question 16

Using dimensional analysis, the period $$T_{osc} of $$oscillation for a ball dropped through a tunnel in the Earth (uniform density) is proportional to:

Accepted Answer

$$T_{osc} \propto \sqrt{\frac{R^3$}{GM}}$$

Question 17

For a satellite in a circular orbit at the Earth's surface (radius $$R$$), what is the orbital period $$T_{orb}$$?

Accepted Answer

$$T_{orb} = 2\pi \sqrt{\frac{R^3$}{GM}}$$

Question 18

Which of the following statements is true about the periods $$T_{osc} ($$oscillation through the Earth) and $$T_{orb} ($$orbital period at Earth's surface)?

Accepted Answer

$$T_{osc} = T_{orb}$$

Newtonian Gravity: Elliptical Orbits and Gravitational Forces

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