Suppose the moon were held in its orbit not by gravity but by a massless cable attached to the center of the earth. What would be the tension in the cable? Use the table of astronomical data inside the back cover of the book.

Verified step by step guidance

1. Identify the necessary values from the astronomical data table, such as the mass of the moon (m), the radius of the moon's orbit (r), and the gravitational constant (G).
2. Use the formula for centripetal force, which is required to keep the moon in its orbit: F_c = \frac{m v^2}{r}. Here, m is the mass of the moon, v is the orbital velocity of the moon, and r is the radius of the orbit.
3. Calculate the orbital velocity (v) of the moon using the formula v = \sqrt{\frac{G M}{r}}, where M is the mass of the Earth and G is the gravitational constant.
4. Substitute the expression for v from step 3 into the formula for centripetal force in step 2. This will allow you to express the centripetal force solely in terms of G, M, m, and r.
5. The tension in the cable would be equal to the centripetal force calculated, as the tension must provide the necessary force to keep the moon in orbit. Thus, Tension (T) = F_c.

Key Concepts

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. In this scenario, the tension in the cable would provide the necessary centripetal force to keep the moon in orbit. The formula for centripetal force is F_c = (mv^2)/r, where m is the mass of the moon, v is its orbital velocity, and r is the radius of the orbit.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. In a typical scenario, the moon is held in orbit by the gravitational pull of the Earth. However, in this hypothetical situation, the gravitational force would not be acting, and the tension in the cable would need to counterbalance the moon's inertia to maintain its circular motion.

Orbital Velocity

Orbital velocity is the speed at which an object must travel to maintain a stable orbit around a celestial body. It depends on the mass of the central body and the radius of the orbit. For the moon, this velocity can be calculated using the formula v = √(GM/r), where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the moon. In this case, the tension in the cable must be sufficient to provide the necessary centripetal force based on this orbital velocity.