Question

Assuming the length of the day uniformly increases by $0.001$ second per century. Calculate the net effect on the measure of time over $20$ centuries.
(A) $3.2$hours
(B) $2.1$hours
(C) $2.4$hours
(D) $5$ hours

Answer 1

Hint Find the increase in length of the day for 20 centuries and then take an average for the increase to find net effect. The increase in length for twenty centuries will be obtained by multiplying the change per century with the no. of centuries. Then one needs to take average for the change in the length of day today and the change in length of the day after twenty centuries. Then convert this time into hours.

Complete Step by Step solution
Length of the day uniformly increases by \[0.001s\]per century.
So, for increase in 20 centuries we get, \[20century\times \dfrac{0.001s}{century}\]
= \[0.02s\]
So, average increase in length for the day today and day after 20 centuries = \[\dfrac{0+0.02}{2}\]
= \[0.01s\]
The increase over 20 centuries is uniform so we can approximate the net effect by considering the average increase per day multiplied by the no. of days in 20 centuries.
So, \[T=\left( \dfrac{0.01s}{day} \right)\times 20centuries\] 1 century = 100 years
\[=\left( \dfrac{0.01s}{day} \right)\times 100\times 20years\]
\[=\left( \dfrac{0.01s}{day} \right)\times 2000years\] 1 year = 365.25 days
\[=\left( \dfrac{0.01s}{day} \right)\times 2000\times 365.25days\]
\[=0.01\times 2000\times 365.25s\]
\[=7305s\]
In hours we get, \[\dfrac{7305}{60\times 60}\]\[=\dfrac{7305}{3600}\]\[\cong 2.03hrs\] 1 hour = 60 min; 1 min = 60 sec
     \[\cong 2.1hrs\]

(Option b) is correct answer

Additional Information It has been observed that over a period of time, the earth’s day has increased in length over time due to tides raised by the moon which slows earth’s rotation. The length of the day at a particular location on earth is a periodic function of time. This is all caused by the \[{{23.5}^{\circ }}\]degree tilt of the earth’s axis as it moves around the sun.
High and low tides are caused by the moon. The Moon’s gravitational pull generates something called the tidal force. The Tidal Force causes Earth and its water to bulge out on the side closest to the moon and the side farthest to from the moon. These bulges of water are high tides.

Note Study the conversions of century into days and hours into seconds. One can read about the concept of tides and the factors causing them. The effect of tides, on the length of the day and other factors includes being the reason for the increase in length of the day over centuries or more.