The rate of rotation of most pulsars gradually decreases because rotational kinetic energy is gradually converted into other forms of energy by a variety of complicated friction process.
Suppose a pulsar of mass 1.5x10^30 kg and radius 20km is spinning at the rate of 2.1 rev/s and is slowing down at the rate of 1x10^-15 rev/s.
A) What is the rate in J/s at which the rotational energy is decreasing?
B) If the rate of decrease is constant, what how long will it take the pulsar to come to a stop?
To calculate the rate at which the rotational energy is decreasing (Question A), we can use the formula for rotational kinetic energy:
Rotational kinetic energy = (1/2) * moment of inertia * angular velocity^2
The moment of inertia of a solid sphere is given by:
Moment of inertia = (2/5) * mass * radius^2
Let's calculate step by step:
1) Calculate the moment of inertia:
Moment of inertia = (2/5) * (mass) * (radius)^2
= (2/5) * (1.5 x 10^30 kg) * (20,000 m)^2
2) Calculate the initial rotational kinetic energy:
Rotational kinetic energy = (1/2) * (moment of inertia) * (angular velocity)^2
= (1/2) * (moment of inertia) * (2.1 rev/s)^2
3) Calculate the final rotational kinetic energy:
Rotational kinetic energy (final) = (1/2) * (moment of inertia) * ((2.1 - 1x10^-15) rev/s)^2
4) Calculate the rate of decrease in rotational energy:
Rate of decrease = (initial rotational kinetic energy) - (final rotational kinetic energy)
Note: Make sure to convert the masses and distances into SI units (kilograms and meters) before performing the calculations.
Now, let's move on to Question B, which asks how long it will take for the pulsar to come to a stop.
If the rate of decrease is constant, then we can use the concept of linear relationships:
Time = Change in quantity / Rate of change
In this case, the "change in quantity" is the initial angular velocity (2.1 rev/s), and the "rate of change" is the given rate of decrease (1x10^-15 rev/s). We can use this equation to calculate the time it will take for the pulsar to come to a stop.