A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period T of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of T 0.033 s that is increasing at the rate of 1.26 105 s/y. (a)What is the pulsar’s angular acceleration a? (b) If a is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year 1054.Assuming constant a, find the initial T.

a) The angular acceleration

= -(2.3045 × 10⁻⁹) rad/s²

b) 2619.1 years from now

c) Initial Period = 0.0208 s

Explanation:

Given, current period of rotation = T = 0.033 s

(dT/dt) = (1.26 × 10⁻⁵) s/y

Converting to SI units, we convert the year to seconds.

(dT/dt) = (1.26 × 10⁻⁵) ÷ (60×60×24×365.25)

(dT/dt) = (3.993 × 10⁻¹³) s/s

a) Angular acceleration is the time derivative of the pulsar's angular velocity

α = (dw/dt)

And the angular velocity is given as

w = (2π/T)

α = (d/dt)(w) = (d/dt)(2π/T)

α = (-2π/T²) (dT/dt)

α = (-2π/0.033²) (3.993 × 10⁻¹³)

α = -(2.3045 × 10⁻⁹) rad/s²

b) Since the angular acceleration is constant, we can use the equation of motion for angular motion.

w = w₀ + αt

w = final angular velocity = 0 rad/s (since it had come to rest at this point)

w₀ = Initial angular velocity (current angular velocity) = (2π/T) = (2π/0.033) = 190.476 rad/s

α = constant angular acceleration

= -(2.3045 × 10⁻⁹) rad/s²

t = time = ?

0 = 190.476 - (2.3045 × 10⁻⁹)t

t = (8.265 × 10¹⁰) s = 2619.1 years from now

c) If the Pulsar originated in 1054, in this year 2020, it has been for

2020 - 1054 = 966 years

Since there's a direct relationship between the period and the number of years that the Pulsar has been

T = Tᵢ + (dT/dt)(t)

T = current period = 0.033 s

Tᵢ = Initial period = ?

(dT/dt) = (1.26 × 10⁻⁵) s/y

t = 966 years

0.033 = Tᵢ + (1.26 × 10⁻⁵ × 966

0.033 = Tᵢ + 0.0121716

Tᵢ = 0.033 - 0.0121716 = 0.0208284 s = 0.0208 s

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