Large stars can explode as they finish burning their nuclear fuel, causing a supernova. The explosion blows away the outer layers of the star. According to Newton’s third law, the forces that push the outer layers away have reaction forces that are inwardly directed on the core of the star. These forces compress the core and can cause the core to undergo a gravitational collapse. The gravitational forces keep pulling all the matter together tighter and tighter, crushing atoms out of existence. Under these extreme conditions, a proton and an electron can be squeezed together to form a neutron. If the collapse is halted when the neutrons all come into contact with each other, the result is an object called a neutron star, an entire star consisting of solid nuclear matter. Many neutron stars rotate about their axis with a period of ≈ 1 s and, as they do so, send out a pulse of electromagnetic waves once a second. These stars were discovered in the 1960s and are called pulsars. How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface?

Verified step by step guidance

1. Step 1: Understand the problem. The question asks for the number of revolutions per minute made by a satellite orbiting 1.0 km above the surface of a neutron star. To solve this, we need to calculate the orbital period of the satellite and then convert it into revolutions per minute.
2. Step 2: Use the formula for the orbital velocity of a satellite: \( v = \sqrt{\frac{GM}{r}} \), where \( G \) is the gravitational constant, \( M \) is the mass of the neutron star, and \( r \) is the orbital radius (sum of the neutron star's radius and the satellite's altitude).
3. Step 3: Calculate the orbital period \( T \) using the relationship \( T = \frac{2\pi r}{v} \). Substitute the expression for \( v \) from Step 2 into this formula to find \( T \).
4. Step 4: Convert the orbital period \( T \) (in seconds) into revolutions per minute. Use the conversion factor \( 1 \text{ revolution} = T \text{ seconds} \) and \( 1 \text{ minute} = 60 \text{ seconds} \). The number of revolutions per minute is given by \( \text{RPM} = \frac{60}{T} \).
5. Step 5: Plug in the known values for \( G \), \( M \), the radius of the neutron star, and the satellite's altitude to calculate \( r \), \( T \), and finally \( \text{RPM} \). Ensure all units are consistent (e.g., meters for distance, seconds for time).

Key Concepts

Newton's Third Law of Motion

Newton's Third Law states that for every action, there is an equal and opposite reaction. This principle is crucial in understanding how forces interact during a supernova explosion, where the outward force from the explosion generates an inward reaction force on the core, leading to gravitational collapse.

Gravitational Collapse

Gravitational collapse occurs when the gravitational forces acting on an object exceed the forces resisting that collapse, leading to a compression of matter. In the context of a supernova, this process can result in the formation of dense objects like neutron stars, as the core's material is crushed under extreme gravitational pressure.

Neutron Stars and Pulsars

Neutron stars are incredibly dense remnants of massive stars that have undergone gravitational collapse, consisting primarily of neutrons. Some neutron stars are observed as pulsars, which emit beams of electromagnetic radiation due to their rapid rotation, creating a pulsing effect as they rotate, similar to a lighthouse beam.