Question

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 \times 10^{- 5} s y^{- 1}

. (a) What is the pulsar's angular acceleration

α

? (b) If

α

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

α

, find the initial

T

.

Accepted Answer

Given:

The period of the pulsar, $T = 0.003 s$

The period increases at a rate of $\frac{d T}{d t} = 1.26 \times 10^{- 5} s y^{- 1}$

(a) A complete revolution is an angular displacement of

$Δ θ = 2 π rad$ , so the angular velocity in $rad s^{- 1}$ is given by:

$ω = \frac{θ}{T} ω = \frac{2 π}{T}$

The angular acceleration is given by:

$α = \frac{d ω}{d t} α = - \frac{2 π}{T^{2}} \frac{d T}{d t}$

For the pulsar detailed in the problem, we have,

$\frac{d T}{d t} = \frac{1.26 \times 10^{- 5} s y^{- 1}}{60 \times 60 \times 24 \times 365.25 s y^{- 1}} \frac{d T}{d t} = 3.992 \times 10^{- 13}$

Therefore,

$α = - [\frac{2 π}{{(0.033 s)}^{2}}] (3.992 \times 10^{- 13}) α = - 2.304 \times 10^{- 9} rad s^{- 2}$

The negative sign denotes that the angular acceleration is opposite to the angular velocity and the pulsar is slowing down.

(b) We solve $ω = ω_{0} + α t$

For the time $t$ , when $ω = 0$ ,

$t = - \frac{ω_{0}}{α} t = - \frac{θ}{α T} t = - \frac{2 π}{α T} t = - \frac{2 π}{(- 2.304 \times 10^{- 9} rad s^{- 2}) (0.033 s)} t = 8.267 \times 10^{10} s$

$t = 2619.8 year$

(c)

$2018 - 1054 = 964 y$

The pulsar is originated in the supernova explosion $964 y$ ago, then the initial period of the pulsar,

$T_{i} = T - 964 \times \frac{d T}{d t} T_{i} = 0.033 - 964 \times 1.26 \times 10^{- 5} T_{i} = 0.012 s$

Important Questions on Rotation

Learn and Prepare for any exam you want !