pulsar is a rapidly rotating neutron star that emits radio pulses with precise synchronization, there being one such pulse for each rotation of the star. The period of rotation is found by measuring the time between pulses. At present, the pulsar in the central region of the Crab nebula has a period of rotation of = 0.09000000 s, and this is observed to be increasing at the rate of 0.00000467 s/yr. What is the angular velocity of the star?What is the angular acceleration of the pulsar?If its angular acceleration is constant, in how many years will the pulsar stop rotating?The pulsar originated in a super-nova explosion in the year A.D. 1054. What was the period of rotation of the pulsar when it was born?
W = 2 pi / T if W is angular velocity.
so W = 2 pi / .09 = 69.8 radians/second
Tin one year - T now = 4.67*10^- 6 seconds
but one year = 3600*24*365 = 3.15*10^7 seconds
so change of T per second = 4.67*10^-6 / 3.15*10^7
= 1.48 *10^-13 seconds/second = dT/dt
Now Either I do some calculus or you convert all that to angular velociteies by brute force.
I have dT/dt
I want dW/dt
but
W = 2 pi/T
dW/dT = 2 pi (-dT/dt)/T^2 =
2 pi (-1.48*10^-13) / (.09)^2
so angular accleration = dW/dt = a
a= - 1148 * 10^-13 radians /s^2
or
a= - 1.15 * 10^-10 radians/s^2
now you have the initial angular velocity and the angular acceleration and the number of seconds per year. I think you can take it from there
To find the angular velocity of the star, we need to convert the period of rotation (T) to the angular velocity (ω). The formula for angular velocity is given by ω = 2π/T.
Using this formula, we can calculate the angular velocity of the star:
ω = 2π/T
= 2π/0.09000000 s
= 69.81317008 rad/s (approximately)
So, the angular velocity of the star is approximately 69.81317008 rad/s.
To find the angular acceleration of the pulsar, we can use the relationship between angular velocity (ω) and angular acceleration (α), given by α = Δω/Δt.
Here, Δω is the change in angular velocity and Δt is the change in time.
Given that the rate of increase in the period of rotation (dT/dt) is 0.00000467 s/yr, we can use this value to calculate the change in angular velocity:
Δω = dΩ/dt = (dω/dt)(Δt)
= (dω/dt)(1 year)
= (0.00000467 s/yr)(2π/0.09000000 s)
= 0.40916667 rad/yr
Therefore, the angular acceleration of the pulsar is approximately 0.40916667 rad/yr.
To calculate how many years it will take for the pulsar to stop rotating, we can use the formula:
Δω = αΔt
Rearranging this formula, we get:
Δt = Δω/α
Substituting the known values, we have:
Δt = 69.81317008 rad/s / 0.40916667 rad/yr
= 170.55884873 years (approximately)
Therefore, if the angular acceleration of the pulsar remains constant, it will take approximately 170.56 years for the pulsar to stop rotating.
To find the period of rotation of the pulsar when it was born, we need to use the concept of conservation of angular momentum. Angular momentum (L) is given by L = Iω, where I is the moment of inertia (which remains constant) and ω is the angular velocity.
Since angular momentum is conserved, we can write:
Iω = I'ω'
where I' and ω' are the moment of inertia and angular velocity at birth, respectively.
Initially, the pulsar had a moment of inertia (I) and angular velocity (ω) at birth. Now, it has a moment of inertia (I') and angular velocity (ω).
By rearranging the equation, we get:
ω' = Iω/I'
To find the initial period of rotation (T'), we can use the formula:
T' = 2π/ω'
Substituting the values, we have:
T' = 2π / (Iω/I')
As we do not have the values of moment of inertia, we cannot calculate the exact initial period of rotation of the pulsar when it was born. This information would require additional data or assumptions.