Question 1

At point B in the elliptical orbit shown, the gravitational force exerted by the Earth on the object can be split into two components: one parallel and one perpendicular to the velocity. What is the effect of the parallel component of the gravitational force on the object's velocity?

Accepted Answer

It increases or decreases the magnitude of the velocity, depending on the direction.

Question 2

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

Accepted Answer

At point A, because it is closest to the Earth and gravitational potential energy is minimum.

Question 3

Given the angular momentum at point A is $$L_A = m v_A r_A$$, and at point C is $$L_C = m v_C r_C$$, what can you conclude about $$L_A$$ and $$L_C in $$the absence of external torques?

Accepted Answer

$$L_A = L_C$$ because angular momentum is conserved.

Question 4

Suppose a rocket thruster at point E exerts a brief, strong force in the direction of motion. What happens to the object's angular momentum about the center of the Earth?

Accepted Answer

It remains unchanged because the force is in the direction of motion.

Question 5

If the rocket at point E pushes the object to escape speed with minimum chemical energy, in which direction should the force be applied?

Accepted Answer

Directly away from the center of the Earth (radially outward).

Question 6

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

Accepted Answer

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

Question 7

Why is the gravitational force on a point mass inside a spherically symmetric shell of mass zero?

Accepted Answer

Because the forces from all parts of the shell cancel due to symmetry.

Question 8

For a point mass $$m at a $$distance $$r$$ from the center of a uniform ball of mass $$M$$ and radius $$R$$, what is the gravitational potential energy $$U_g(r)$$ for $$r \u003e R$$?

Accepted Answer

$$U_g(r) = -G \frac{M m}{r}$$

Question 9

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

Accepted Answer

Simple harmonic motion.

Question 10

What is the effective 'spring constant' $$k$$ for the gravitational force acting on a mass $$m$$ inside the Earth (radius $$R$$, mass $$M$$)?

Accepted Answer

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

Question 11

Using dimensional analysis, the period $$T_{osc} of $$the ball’s oscillatory motion through the Earth is independent of which quantity?

Accepted Answer

The mass of the ball $$m$$

Question 12

If the ball were instead orbiting the Earth at the radius of the Earth, what would be its orbital period $$T_{orb} in $$terms of $$G$$, $$M$$, and $$R$$?

Accepted Answer

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

Question 13

The gravitational field $$\vec{g}$$ due to a point mass $$M at $$the origin is given by:

Accepted Answer

$$\vec{g} = -\hat{r} G M / r^2$

Question 14

If a rocket at point E increases the object's 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

$$\alpha = \sqrt{1 + \frac{2 G M}{r_E v_E^2$}}$$

Question 15

Why does it require more chemical energy to achieve escape speed if the rocket thruster is not pointed radially outward?

Accepted Answer

Because only the radial component of the force increases the total energy efficiently.

Question 16

The gravitational potential $$\Phi$$ for a point mass $$M at $$the origin is:

Accepted Answer

$$\Phi = -G M / r$$

Question 17

Which of the following best describes the relationship between the gravitational force and the gradient of the gravitational potential energy $$U_g$$?

Accepted Answer

$$\vec{F}_g = -
abla U_g$$

Newtonian Gravity: Elliptical Orbits, Forces, and Energetics

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