Time standards are now based on atomic clocks. A promising second standard is based on pulsars, which are rotating neutron stars (highly compact stars consisting only of neutrons). Some rotate at a rate that is highly stable, sending out a radio beacon that sweeps briefly across Earth once with each rotation, like a lighthouse beacon. Suppose a pulsar rotates once every 1.596 806 448 872 75 4 ms, where the trailing 4 indicates the uncertainty in the last decimal place (it does not mean 4 ms).

(a) How many times does the pulsar rotate in 21.0 days?

(b) How much time does the pulsar take to rotate 3.0 106 times? (Give your answer to at least 4 decimal places.)
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(c) What is the associated uncertainty of this time?
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I will be happy to critique your thinking.

To answer these questions, we need to use the given information about the rotation rate of the pulsar.

(a) To find out how many times the pulsar rotates in 21.0 days, we first need to convert the length of time from days to milliseconds. There are 24 hours in a day, 60 minutes in an hour, 60 seconds in a minute, and 1,000 milliseconds in a second. Therefore, the conversion factor is:

1 day = 24 hours/day * 60 minutes/hour * 60 seconds/minute * 1,000 milliseconds/second

Converting 21.0 days to milliseconds:

21.0 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute * 1,000 milliseconds/second = 1,814,400,000 milliseconds

Next, we divide the total number of milliseconds by the rotation period of the pulsar:

1,814,400,000 milliseconds / 1.596 806 448 872 75 4 ms = 1,136,417,361.154 rotations

Therefore, the pulsar rotates approximately 1,136,417,361 times in 21.0 days.

(b) To calculate the time it takes for the pulsar to rotate 3.0 × 10^6 times, we multiply the rotation period by the number of rotations:

3.0 × 10^6 rotations * 1.596 806 448 872 75 4 ms = 4,790.4192 milliseconds

So, the pulsar takes approximately 4,790.4192 milliseconds to complete 3.0 × 10^6 rotations.

(c) The associated uncertainty in time is given as the trailing 4 in the rotation period, which signifies the uncertainty in the last decimal place. Therefore, the uncertainty is 0.0001 milliseconds (as each digit represents ten times less precision than the previous one).

Thus, the uncertainty in time is 0.0001 seconds.