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 vector. 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 orbit (A, B, C, D, or E) is the object's speed the greatest, and why?

Accepted Answer

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

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 say about $$L_A$$ and $$L_C if $$only gravity acts on the object?

Accepted Answer

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

Question 4

If 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

Suppose the rocket at point E pushes the object to escape speed. In which direction should the thruster push to minimize the required chemical energy?

Accepted Answer

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

Question 6

The gravitational force on a point mass inside a spherically symmetric shell of mass is zero. Which argument best explains this result?

Accepted Answer

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

Question 7

For a point mass $$m$$ outside a uniform spherical ball of mass $$M$$ and radius $$R$$, what is the gravitational force acting on $$m at $$distance $$r \u003e R$$?

Accepted Answer

$$F = -\frac{G M m}{r^2$}$$

Question 8

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

Accepted Answer

Simple harmonic motion.

Question 9

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

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

Question 10

The period of oscillation $$T_{osc}$$ for a ball dropped through a tunnel in the Earth is independent of the ball's mass. Which principle does this illustrate?

Accepted Answer

The equivalence principle (Galileo's principle).

Question 11

If the ball instead orbits the Earth at radius $$R$$, what is the orbital period $$T_{orb}$$? (Assume $$M is $$Earth's mass, $$R is $$Earth's radius, $$G is $$the gravitational constant.)

Accepted Answer

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

Question 12

The gravitational potential energy $$U_g(r)$$ for a point mass $$m$$ outside a uniform sphere of mass $$M is$$:

Accepted Answer

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

Question 13

Why does the gravitational force exerted by a spherically symmetric shell on a point mass outside the shell act as if all the mass were concentrated at the center?

Accepted Answer

Because the net force from each shell element adds up to a radial force equivalent to a point mass at the center.

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 $$M ($$Earth's mass), $$r_E ($$distance from Earth's center), and $$v_E$$?

Accepted Answer

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

Question 15

The gravitational field $$\vec{g} at a $$distance $$r$$ from a point mass $$M is$$:

Accepted Answer

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

Question 16

If the rocket at point E pushes in a direction other than radially outward, why does it require more chemical energy to achieve escape speed?

Accepted Answer

Because only the radial component of the velocity contributes to escaping Earth's gravity.

Question 17

Which of the following best describes Kepler's second law as it applies to the orbit shown?

Accepted Answer

The object sweeps out equal areas in equal times, so the rate at which area is swept out is constant.

Newtonian Gravity: Orbits, Forces, and Energies – Step-by-Step Study Guidance

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드