A star the size of our Sun runs out of nuclear fuel and, without losing mass, collapses to a white dwarf star the size of our Earth. The radius of our Sun is 6.96×108m , the radius of Earth is 6.37×106m .
If the star initially rotates at the same rate as our Sun, which is once every 25 days, determine the rotation rate of the white dwarf.
angular momentum remains the same.
momentinertia*w=finalmomentinertia*wf
2/5(mri^2)wi=2/5(mrf^2)wf
wf=wi*(ri/rf)^2
wf=2pi/25 *(6.96E8/6.37E6)^2 rad/day
wf=3000 rad/day or 477 rotations/day
check the math.
mastering physics requires the answer to be in rad/sec so the answer ends up being .035 rad/s
To determine the rotation rate of the white dwarf, we can use the principle of conservation of angular momentum.
The angular momentum of an object can be calculated by multiplying its moment of inertia by its angular velocity. Since the star does not lose mass during the collapse, its moment of inertia remains the same.
The moment of inertia of a solid sphere is given by the formula:
I = (2/5) * m * r^2
Where:
I = moment of inertia
m = mass of the object
r = radius of the object
Now, let's calculate the initial moment of inertia of the star.
Given:
Radius of the star (initially) = 6.96 × 10^8 meters (R_star)
Radius of the white dwarf = 6.37 × 10^6 meters (R_wd)
Rotation period of the star (initially) = 25 days = 25 * 24 * 60 * 60 seconds (T_star)
The mass of the star remains the same, so we can ignore it in the calculations.
Moment of inertia of the star (initially):
I_star = (2/5) * m * R_star^2
Now, let's calculate the moment of inertia of the white dwarf.
Moment of inertia of the white dwarf:
I_wd = (2/5) * m * R_wd^2
Since the star does not lose mass, we can assume the mass (m) remains constant. Therefore, the ratio of the moment of inertia between the star and the white dwarf is:
I_star / I_wd = (R_star^2) / (R_wd^2)
Additionally, we are given that the initial rotation period of the star (T_star) is 25 days.
The rotation period of an object is inversely proportional to its angular velocity. Therefore, the ratio of the rotation period between the star and the white dwarf is:
T_star / T_wd = omega_wd / omega_star
where:
T_wd = rotation period of the white dwarf (what we want to calculate)
omega_wd = angular velocity of the white dwarf
omega_star = angular velocity of the star (initially)
We can equate these two ratios:
[(R_star^2) / (R_wd^2)] = (omega_wd / omega_star)
Now we can solve for the rotation period of the white dwarf. Rearranging the equation:
omega_wd = omega_star * [(R_star^2) / (R_wd^2)]
T_wd = T_star / [(R_star^2) / (R_wd^2)]
Substituting the given values:
T_wd = (25 * 24 * 60 * 60) / [(6.96 × 10^8)^2 / (6.37 × 10^6)^2]
By evaluating this expression, we can determine the rotation rate of the white dwarf.
To determine the rotation rate of the white dwarf, we need to understand the concept of conservation of angular momentum. Angular momentum describes the rotational motion of an object and is given by the equation:
L = Iω,
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
When a star collapses into a white dwarf, it undergoes a significant decrease in size, resulting in a decrease in moment of inertia (I). Since the star does not lose mass, we can assume that the moment of inertia remains constant.
Now, we can set up an equation to establish the conservation of angular momentum before and after the collapse. Let's assume the initial angular velocity of the star is ω_initial, and the final angular velocity of the white dwarf is ω_final.
Since angular momentum is conserved, we have:
L_initial = L_final
I_initial * ω_initial = I_final * ω_final
Given that the initial star rotates once every 25 days, we can convert this to angular velocity using the relation:
ω_initial = (2 * π) / T_initial,
where T_initial is the initial rotation period in seconds.
Similarly, we can express the rotation period of the white dwarf using:
ω_final = (2 * π) / T_final,
where T_final is the rotation period of the white dwarf in seconds.
Substituting these values into the conservation equation, we get:
I_initial * (2 * π) / T_initial = I_final * (2 * π) / T_final
Simplifying this equation, we find:
ω_final = (I_initial / I_final) * (T_final / T_initial) * ω_initial
To solve for the rotation period of the white dwarf, we need to determine the ratio of the initial and final moments of inertia (I_initial / I_final). Since the star collapses to a white dwarf with the same mass but a significantly smaller radius, we can use the formula for the moment of inertia of a solid sphere:
I = (2/5) * M * R^2,
where I is the moment of inertia, M is the mass, and R is the radius.
For the star initially and the white dwarf finally, we have:
I_initial = (2/5) * M_initial * R_initial^2
I_final = (2/5) * M_final * R_final^2
Since the mass does not change during the collapse, we can cancel it out in the equation.
After substituting the values for the radii (R_initial = 6.96 × 10^8 m and R_final = 6.37 × 10^6 m), we can calculate the ratio of the moments of inertia (I_initial / I_final).
Finally, we can plug this ratio, along with the initial and final rotation periods, into the equation for ω_final to find the rotation rate of the white dwarf.