Question

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 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

1. The speed of a particle on the star’s equator is$\mathrm{v}=1.3\times {10}^5\mathrm{m}/\mathrm{s}$
2. The magnitude of the particle’s centripetal acceleration is$\mathrm{a}=7.9\times {10}^5\mathrm{m}/{\mathrm{s}}^2$
3. If the rotation of a neutron star increases, the speed of a particle on the star’s equator and the magnitude of the particle’s centripetal acceleration increase.

Step-by-step solution

Given

The radius of star is,

$$\begin{gathered} \mathrm{r}=20\mathrm{km} \\ =20\times {10}^3\mathrm{m} \end{gathered}$$

The period of the star is T = 1.00 s

Understanding the concept

A neutron star is rotating once at every second. The particle is placed on the star’s equator; hence it covers the distance as circumference of circle within time T. We can use the formula of speed and centripetal acceleration and find the effect of faster rotation of the neutron star on the speed and centripetal acceleration of the particle.

Formula:

The speed of a particle on a star’s equator is,

$v=\frac{2\mathrm{\pi r}}{T}$ …(i)

The centripetal acceleration is,

$a=\frac{v^2}{r}$ …(ii)

(a) Calculate the speed of a particle on the star’s equator

Use equation (i) to calculate the speed of the particle.

$$\begin{gathered} \mathrm{v}=\frac{2\mathrm{\tau \tau r}}{\mathrm{T}} \\ =\frac{2\mathrm{\tau \tau }\times 20\times {10}^3\mathrm{m}}{1.00\mathrm{s}} \\ =1.3\times {10}^5\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, the speed of the particle on the star’s equator is $1.3\times {10}^5\mathrm{m}/\mathrm{s}$.

(b) Calculate the magnitude of the particle’s centripetal acceleration

Use equation (ii) to calculate the magnitude of the particle’s centripetal acceleration.

$$\begin{gathered} \mathrm{a}=\frac{{\mathrm{v}}^2}{\mathrm{r}} \\ =\frac{{(1.3\times {10}^5\mathrm{m}/\mathrm{s})}^2}{20\times {10}^3\mathrm{m}} \\ =7.9\times {10}^5\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$$7.9\times {10}^5\mathrm{m}/{\mathrm{s}}^2$

Therefore, the magnitude of the particle’s centripetal acceleration is $7.9\times {10}^5\mathrm{m}/{\mathrm{s}}^2$.

(c) Figure out if the answers to (a) and (b) increase, decrease, or remain the same in case the neutron star rotates faster

If the rotation of neutron star increases, its period decreases. Hence, the speed of a particle on star’s equator increases. The magnitude of the particle’s centripetal acceleration depends on speed of the particle; hence,this increasing also increases centripetal acceleration.