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Chapter 6: Problem 31

Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun, but a much smaller diameter. If you weigh 675 \(\mathrm{N}\) on the earth, what would you weigh on the surface of a neutron star that has the same mass as our sun and a diameter of 20.0 \(\mathrm{km} ?\)

Short Answer

Expert verified
You would weigh approximately \( 9.07 \times 10^{12} \text{N} \) on the neutron star.

Step by step solution

01

Calculate Earth's Gravitational Acceleration

On Earth, the gravitational acceleration \( g \) is approximately \( 9.8 \, \text{m/s}^2 \). This means your weight of 675 N corresponds to your mass under Earth's gravity. We need to calculate the mass \( m \) using \( m = \frac{W}{g} \), where \( W \) is the weight.
02

Determine Your Mass

We can find your mass by rearranging the weight equation: \( m = \frac{675 \, \text{N}}{9.8 \, \text{m/s}^2} \). Thus, \( m \approx 68.88 \, \text{kg} \).
03

Calculate Gravitational Force on Neutron Star

For the neutron star, the gravitational force equation \( F = \frac{G M m}{r^2} \) applies. Here, \( G \) is the gravitational constant, \( M \) is the mass of the neutron star, and \( r \) is the radius of the star. The mass of the neutron star is the same as the sun, \( M = 1.989 \times 10^{30} \text{kg} \).
04

Convert Diameter to Radius

The problem gives the diameter of the neutron star as 20.0 km, so the radius \( r \) is half of this: \( r = 10.0 \text{km} \) or \( 10,000 \text{meters} \).
05

Apply Gravitational Formula

Now, use the equation \( F = \frac{(6.674 \times 10^{-11} \text{m}^3\text{kg}^{-1}\text{s}^{-2}) (1.989 \times 10^{30} \text{kg}) (68.88 \text{kg})}{(10,000 \text{m})^2} \) to calculate the weight on the neutron star.
06

Compute the Result

Perform the calculations to find \( F = 9.07 \times 10^{12} \, \text{N} \). This would be your weight on the surface of the neutron star.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Neutron Stars
Neutron stars are fascinating celestial objects that form when a massive star dies in a supernova explosion. While they have the mass of the sun, they are incredibly dense and have a small radius, typically only around 10 kilometers. This compact size is astonishing when compared to the vast dimensions of our sun.
  • Densely Packed: A neutron star's matter is packed so tightly that a sugar-cube-sized amount of neutron-star material would weigh about as much as all of humanity.
  • Gravitational Effect: The high density means they have intense gravitational fields compared to Earth or even larger celestial bodies.
  • Pulsars: Some neutron stars emit beams of electromagnetic radiation, called pulsars due to the way these beams sweep across space.
Neutron stars' extreme gravity and compactness make them perfect laboratories for studying strong gravitational fields and the physics of nuclear density.
Mass and Weight
Mass and weight are two interrelated yet distinct concepts in physics. The mass of an object refers to the amount of matter it contains and is measured in kilograms. It remains constant regardless of the object's location in the universe.
On the other hand, weight measures the force exerted on that mass by gravity and is dependent on the gravitational field strength. Hence, weight can vary based on where the object is located.
  • Constant Mass: If you are 68.88 kg on Earth, you'll remain 68.88 kg on the moon, a neutron star, or wherever you go.
  • Variable Weight: As gravity changes, so does your weight. That's why you weigh less on the moon compared to Earth, due to the moon鈥檚 weaker gravity.
  • Equation: Weight can be calculated as the product of mass and the acceleration due to gravity, i.e., \( W = mg \).
Understanding the difference between mass and weight is essential when dealing with problems involving gravitational forces.
Gravitational Acceleration
Gravitational acceleration refers to the rate at which objects accelerate towards one another due to the force of gravity. On Earth's surface, this acceleration is a familiar constant, roughly \(9.8 \, \text{m/s}^2\). This value helps us calculate an object's weight, given its mass.
When dealing with neutron stars, however, the gravitational acceleration becomes immensely stronger due to their incredible density and mass compactness.
  • On Earth: Gravity gives you a weight based on Earth鈥檚 gravitational acceleration of \(9.8 \, \text{m/s}^2\).
  • On Neutron Stars: Due to their tiny radius yet massive size, gravitational acceleration can skyrocket, making even simple objects incredibly heavy on the surface.
  • The Formula: The formula \( F = \frac{G M m}{r^2} \) is used to calculate the gravitational force, where \(G\) is the universal gravitational constant, \(M\) is the mass of the celestial body, \(m\) is the mass of the object, and \(r\) is the distance from the center to the surface.
Gravitational acceleration explains why objects weigh different amounts on different planets or stars, translating the mass's consistent nature into varying weights based on local gravitational fields.

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Most popular questions from this chapter

A bowling ball weighing 71.2 \(\mathrm{N}\) is attached to the ceiling by a 3.80 \(\mathrm{m}\) rope. The ball is pulled to one side and released; it then swings back and forth like a pendulum. As the rope swings through its lowest point, the speed of the bowling ball is measured at 4.20 \(\mathrm{m} / \mathrm{s}\) . At that instant, find (a) the magnitude and direction of the acceleration of the bowling ball and (b) the tension in the rope. Be sure to start with a free-body diagram.

Artificial gravity in space stations. One problem for humans living in outer space is that they are apparently weight- less. One way around this problem is to design a cylindrical space station that spins about an axis through its center at a constant rate. (See Figure \(6.32 . )\) This spin creates "artificial gravity" at the outside rim of the station. (a) If the diameter of the space station is \(800.0 \mathrm{m},\) how fast must the rim be moving in order for the "artificial gravity" acceleration to be \(g\) at the outer rim? (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface. How fast must the rim move in this case? (c) Make a free-body diagram of an astronaut at the outer rim.

The Indy 500 . The Indianapolis Speedway (home of the Indy 500 ) consists of a 2.5 mile track having four turns, each 0.25 mile long and banked at \(9^{\circ} 12^{\prime} .\) What is the no-friction-needed speed (in \(\mathrm{m} / \mathrm{s}\) and mph) for these turns? (Do you think drivers actually take the turns at that speed?

Huygens probe on Titan. In January 2005 the Huygens probe landed on Saturn's moon Titan, the only satellite in the solar system having a thick atmosphere. Titan's diameter is \(5150 \mathrm{km},\) and its mass is \(1.35 \times 10^{23} \mathrm{kg}\) . The probe weighed 3120 \(\mathrm{N}\) on the earth. What did it weigh on the surface of Titan?

The asteroid 234 Ida has a mass of about \(4.0 \times 10^{16} \mathrm{kg}\) and an average radius of about 16 \(\mathrm{km}\) (it's not spherical, but you can assume it is). (a) Calculate the acceleration of gravity on 234 Ida. (b) What would an astronaut whose earth weight is 650 \(\mathrm{N}\) weigh on 234 lda? (c) If you dropped a rock from a height of 1.0 \(\mathrm{m}\) on 234 Ida, how long would it take to reach the ground? (d) If you can jump 60 \(\mathrm{cm}\) straight up on earth, how high could you jump on 234 Ida? (Assume the asteroid's gravity doesn't weaken significantly over the distance of your jump.)
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