If 2 planets are aligned and they circle around a star and the first planet takes 7 days to make one revolution and the second planet takes 11 days, how many days in hours, minutes and seconds do they take to align themselves again.

2. Originally Posted by justincase
If 2 planets are aligned and they circle around a star and the first planet takes 7 days to make one revolution and the second planet takes 11 days, how many days in hours, minutes and seconds do they take to align themselves again.
Angular speed of planet #1, w1 = 1rev /7days = 2pi/7 rad/day
Angular speed of planet #2, w2 = 2pi/11 rad/day.

Planet #1, being faster, should have revolved once before planet #2 finishes its first revolution. Or, let us assume that in the same amount of time, the two planets will be alligned again such that the faster one is ahead by one revolution.

Angular distance, or angle, = (angular speed)*(time)

In t days time,
Planet #1 covers (w1)*t
Planet #2 covers (w2)*t
But,
(w1)*t -(w2)*t = 1 revolution = 2pi
So,
(2pi/7)*t -(2pi/11)*t = 2pi
(1/7)t -(1/11)t = 1
(4/77)t = 1
t = 77/4 = 19.25 days

That means, in 19.25 days the two planets will be aligned again.

Check:

In 19.25 days,
>>>planet #1, revolving once every 7 days, should have gone (2 +5.25/7) revolutions....or 2.75 revs.
>>>planet #2, at 11 days per rev, should have gone (1 +8.25/11) revolutions ....or 1.75 revs

That means the faster planet has completed more than 2 revs before it was able to catch up with the slower planet. It looks like our assumption is wrong.
But then, the slower planet has completed more that 1 rev when it was caught up. Meaning, between the two planets, there is only 1 rev difference. So our assumption is still correct.

Therefore, after 19 days and 6 hours, the two planets are again aligned. ----answer.

3. Hello, justincase!

A similar approach . . .

If two planets are aligned and they circle around a star
and the first planet takes 7 days to make one revolution
and the second planet takes 11 days,
in how many days do they align themselves again?
I must assume that the measurements are taken in our "days".

Planet $A$ take 7 days to travel 360°.
. . It travels: $\frac{360}{7}$ degrees per day.
.In $x$ days, it travels: $\frac{360}{7}x$ degrees.

Planet $B$ takes 11 days to travel 360°.
. . It travels: $\frac{360}{11}$ degrees per day.
. . In $x$ days, it travels: $\frac{360}{11}x$ degrees.

They start together, and $A$ (the faster planet) races ahead of $B.$

When they are together again, $A$ has gone one complete revolution, plus $B$'s distance.

The equation is: . $\frac{360}{7}x \;=\;\frac{360}{11}x + 360$

Multiply by $\frac{77}{360}\!\:\;\;11x + 7x + 77 \quad\Rightarrow\quad x \:=\:\frac{77}{4}$

Therefore, they are aligned again in 19¼ days = 19 days, 6 hours.