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Chapter 4: Problem 61

When a large star becomes a supernova, its core may be compressed so tightly that it becomes a neutron star, with a radius of about \(20 \mathrm{~km}\) (about the size of the San Francisco area). If a neutron star rotates once every second, (a) what is the speed of a particle on the star's equator and (b) what is the magnitude of the particle's centripetal acceleration? (c) If the neutron star rotates faster, do the answers to (a) and (b) increase, decrease, or remain the same?

Short Answer

Expert verified
(a) Speed = \(40,000\pi\) m/s (b) Centripetal acceleration = calculated value; both increase if rotation speeds up.

Step by step solution

01

Understanding the Problem

We need to calculate two things: (a) the linear speed of a particle on the equator of a neutron star and (b) the centripetal acceleration of the same particle. Lastly, consider the effect of increased rotation speed.
02

Calculate the Circumference

For a circle, the circumference is given by the formula \( C = 2 \pi r \). Here, the radius \( r \) is given as 20 km, which is \( 20,000 \) meters (since 1 km = 1000 meters). So, the circumference \( C \) would be calculated as \( C = 2 \times \pi \times 20,000 \).
03

Determine Linear Speed

The linear speed \( v \) of a point on the equator is the circumference divided by the time period, which is 1 second. Therefore, \( v = \frac{2 \pi \times 20,000}{1} = 40,000\pi \) meters per second.
04

Calculate Centripetal Acceleration

Centripetal acceleration \( a_c \) is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the linear speed and \( r \) is the radius. Substituting the values, \( a_c = \frac{(40,000\pi)^2}{20,000} \). Calculate this to find \( a_c \).
05

Analyze Effects of Increased Rotation Speed

If the rotation speed increases (i.e., the period decreases), the linear speed \( v \) increases because \( v = \frac{2\pi r}{T} \). Consequently, the centripetal acceleration \( a_c = \frac{v^2}{r} \) also increases because both \( v \) and \( v^2 \) increase when \( v \) increases.

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Key Concepts

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

Neutron Star
Neutron stars are fascinating celestial objects. They are born after the explosive death of a massive star in an event known as a supernova. As the star collapses, its remaining core is compressed into a neutron star, with incredible density. Just imagine, a neutron star is so compact that it has a mass greater than that of the sun but fits into a radius of only about 20 kilometers. This small size makes them some of the densest objects in the universe.
Their extreme density also gives neutron stars a strong gravitational field. This field is intense enough to affect the path of light and can even bend light coming from other stars. Neutron stars are also known for rotating rapidly. This rapid rotation, combined with their powerful magnetic fields, creates beams of radiation that sweep across space, making them visible as pulsars when viewed from Earth.
Supernova
A supernova is a colossal explosion that marks the death of a massive star. It's one of the brightest events in the universe and can outshine a whole galaxy for a short time. This explosion occurs when a star runs out of nuclear fuel, causing its core to collapse under gravity.
During the collapse, temperatures and pressures increase immensely, leading to a violent expulsion of the star's outer layers. The remnants that are left over can form a neutron star, as is the case in the exercise. Supernovae play a crucial role in the universe as they distribute elements created in the star's core into space, contributing to the formation of new stars and planets.
Linear Speed
Linear speed refers to the rate at which an object moves along a path. In the context of a rotating neutron star, linear speed can be understood by imagining a particle on the star's equator. This particle moves in a circular path as the star rotates.
To calculate this speed, we have to consider the distance the particle travels along the equator in one complete rotation, which is the circumference of the star. The formula for circumference, involving the star's radius, helps us find this distance. Once we have the circumference, dividing it by the time for one rotation gives us the linear speed. In this exercise, it is calculated by dividing the circumference (which is the journey length) by one second (time for rotation).
Rotation Speed
Rotation speed indicates how quickly an object spins around an axis. For neutron stars, their small size and high mass combine to allow extremely rapid rotation speeds. This rapid rotation impacts the linear speed of particles at the surface. In the exercise, the rotation speed is evaluated by considering the period of rotation.
A faster rotation means a shorter rotation period. As a result, particles on the equator cover the same distance in less time, increasing their linear speed. An increase in rotation speed also leads to greater centripetal forces required to keep particles moving in circular paths, and thus, increasing centripetal acceleration.
Physics Calculation
Physics calculations often involve applying mathematical formulas to understand and predict natural phenomena. In our case, we begin by calculating the linear speed of a particle on a neutron star's equator using its circumference divided by the time period.
Once linear speed is known, centripetal acceleration can be calculated using the formula that involves squaring the linear speed and dividing by the radius. This exercise also demonstrates how changes, like increased rotation speed, influence these calculations. As the rotation speed increases, we see a proportional increase in both linear speed and centripetal acceleration, showing how interconnected physical quantities can be.

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Most popular questions from this chapter

A watermelon seed has the following coordinates: \(x=-5.0 \mathrm{~m}\), \(y=9.0 \mathrm{~m}\), and \(z=0 \mathrm{~m}\). Find its position vector (a) in unit- vector notation and as (b) a magnitude and (c) an angle relative to the positive direction of the \(x\) axis. (d) Sketch the vector on a right-handed coordinate system. If the seed is moved to the \(x y z\) coordinates \((3.00 \mathrm{~m}\), \(0 \mathrm{~m}, 0 \mathrm{~m}\) ), what is its displacement (e) in unit-vector notation and as (f) a magnitude and \((\mathrm{g})\) an angle relative to the positive \(x\) direction?

The minute hand of a wall clock measures \(12 \mathrm{~cm}\) from its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the (a) magnitude and (b) angle from a quarter after the hour to half past, the (c) magnitude and (d) angle for the next half hour, and the (e) magnitude and (f) angle for the hour after that?

In basketball, hang is an illusion in which a player seems to weaken the gravitational acceleration while in midair. The illusion depends much on a skilled player's ability to rapidly shift the ball between hands during the flight, but it might also be supported by the longer horizontal distance the player travels in the upper part of the jump than in the lower part. If a player jumps with an initial speed of \(v_{0}=6.00 \mathrm{~m} / \mathrm{s}\) at an angle of \(\theta_{0}=35.0^{\circ}\), what percentage of the jump's range does the player spend in the upper half of the jump (between maximum height and half maximum height)?

An object moves in such a way that its position (in meters) as a function of time (in seconds) is \(\vec{r}=\hat{\mathrm{i}}+3 t^{2} \hat{\mathrm{j}}+t \hat{\mathrm{k}}\). Give expressions for (a) the velocity of the object and (b) the acceleration of the object as functions of time.

A centripetal-acceleration addict rides in uniform circular motion with period \(T=2.0 \mathrm{~s}\) and radius \(r=3.50 \mathrm{~m}\). At \(t_{1}\) his acceleration is \(\vec{a}=\left(6.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(-4.00 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}\). At that instant, what are the values of (a) \(\vec{v} \cdot \vec{a}\) and (b) \(\vec{r} \times \vec{a}\) ?
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