One gear pulley completes one revolution every 3 seconds; another gear pulley…
2023
One gear pulley completes one revolution every 3 seconds; another gear pulley completes one revolution every 5 seconds. If both start together, after how many seconds will they be together again?
- A.
10 sec
- B.
12 sec
- C.
15 sec
- D.
19 sec
Attempted by 19 students.
Show answer & explanation
Correct answer: C
Concept
When two rotating bodies start together and each one returns to its own starting position after a fixed period (the time for one full revolution), the two are next together again -- both simultaneously back at the start -- at a time equal to the LCM (Least Common Multiple) of their two periods, since that is the first instant common to both individual return-times.
Application
The first gear pulley has a period of 3 seconds, so it is back at its starting position at t = 3, 6, 9, 12, 15, 18, ... seconds.
The second gear pulley has a period of 5 seconds, so it is back at its starting position at t = 5, 10, 15, 20, ... seconds.
The smallest time common to both lists is the LCM of 3 and 5: LCM(3, 5) = 15.
So both gear pulleys are back at their starting position together for the first time at t = 15 seconds.
Cross-check
At t = 15 s, the first gear pulley has completed 15 / 3 = 5 whole revolutions and the second has completed 15 / 5 = 3 whole revolutions -- both whole numbers, confirming both are exactly at their starting position. No smaller positive common multiple of 3 and 5 exists, so 15 seconds is the earliest such time.
Correct answer: 15 sec