A circular gear with a diameter of 24 centimeters is mounted directly

Question

A circular gear with a diameter of 24 centimeters is mounted directly on another circular gear with a diameter of 96 centimeters. Both gears turn on the same axle at their exact centers and each gear has a single notch, at the 12 o'clock position. At the same moment, the gears begin to turn at the same rate, with the larger gear moving clockwise and the smaller gear counterclockwise. How far, in centimeters, will the notch on the larger gear have traveled the second time the notches pass each other?

(A) \(32.2 \pi\)

(B) \(35.6 \pi\)

(C) \(38.4 \pi\)

(D) \(39.2 \pi\)

(E) \(40.8 \pi\)

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(A)  32.2 \pi

(B)  35.6 \pi

(C)  38.4 \pi

(D)  39.2 \pi

(E)  40.8 \pi

mikemcgarry · Accepted Answer

I'm happy to help with this. :-)

For finding when they pass each other, the sizes don't matter, because only rotational rate matters. One turns clockwise, one counterclockwise. They start at the same place, 12 o'clock position. When they each turn 180 degrees, they pass the first time at the 6 o'clock position. Then they each travel another 180 degrees, they pass each other a second time back at the 12 o'clock position. Thus, at the second passing, each one has made a full 360 rotation. How far in centimeters will the notch on the larger (d = 96 cm) gear travel in one 360-degree revolution? Well, that's one circumference ===> c = (pi)*d = 96*pi centimeters. This is not listed as an answer, so something is badly messed up about this question. Either the answers listed do not include the correct answer, due to a typo, or the way the question is phrased does not accurately capture what the question writer was trying to say. Either way, this is a faulty question. (If getting it wrong made you doubt yourself, take heart --- it's not you!) Does all all this make sense? Mike :-)

MBArenaissance · Answer

I beg to (politely differ) from Mike....

I believe the answer is C.

My explanation: the size does matter because by the time the smaller gear (circumference 24*pi) goes full circle(i.e 360degrees), it wud have moved a distance of 24*pi....
and the larger gear, by moving the same distance of 24*pi wud have only gone a quarter of its total distance (i.e a quarter of 96*pi)...

in a nut shell, by moving a distance of 24*pi, smaller gear has gone full circle, so its back at 12oclock and larger gear has only just covered a quarter of its whole distance. So they must have met somewhere in the 1st quadrant. (FIRST MEETING POINT)

by the time the smaller gear has gone another half circle, i.e a distance of 12*pi more, it will be positioned at the 6oclock point.. and it wud have covered 36*pi distance in all...
and for the larger gear to have covered 36*pi in distance it'll will only be at the mid point, around 135 degrees... so because the smaller gear can cover its track faster than the larger gear (cos of its shorter distance), their SECOND MEETING POINT will be closer to the current position of the larger gear (just a little after its 135 degree mark).

And looking thru the answer choices, that'll be somewhere around the 38.4*pi mark..... and NOT the 41.6*pi mark.... So answer will be A...

P.S Mike, ur product (Magoosh) is amazing... Currently on the 1week trial... I just might upgrade...

mikemcgarry · Answer

Dear MBArenaissance , Ah, good work. I had no idea where they were getting their answers, but now I have a very focused idea of the problem with the question. The problem comes down to this phrase: the gears start to rotate at the same rate What does " at the same rate " mean? We could talk about either (a) a linear speed , called the tangential velocity in Physics, as MBArenaissance and as, apparently, the question author intended. or (b) a rotation speed , called the angular velocity or angular frequency in Physics. This was my interpretation. I would argue that, if by " same rate " they intend "same linear speed", that needs to be stated unambiguously as part of the problem. I would argue that in most context in which the phrase " rotate at the same rate " is used, the general assumption is that the rotational speeds, not the linear speeds, are equal. That most certainly would be the assumption in a Physics context. One thing that would have made the problem much clearer is if, instead of being coaxial (a strange configuration for gears!), the gears were actually on the same plane and interlocked the way gears are supposed to be. That would guarantee equal tangential velocities in an unambiguous way. Again, I would say that if anyone approaches this question and is utterly befuddled by the choices available, that's not the fault of the test taker. That's the fault of some rather shoddy phrasing in this question. Mike :-)

SVaidyaraman · Answer

The need to make the right assumption is what makes this question difficult. If the assumption that the number of rotations per minute be equal is made, then there is practically nothing to solve. Also there are no choices that correspond to this assumption. Considering the difficulty level of the question and also considering the fact that there are no choices that support the equal rotations per minute assumption, I think the student has to make the assumption that the linear speed only is the same.

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12bhang · Answer

Thank you for your responses. 
But, I cannot understand two things:

A) The questions states that the gears are mounted on the same axle , then how can one turn clockwise and the other anti co
lcokwise?

B) What MBAintern said makes sense when the gears are meshing. Then the linear velocity will be constant and the angular velocity will be inversely proportional to the dia.

i.e N2/N1=D2/D1

Then, I suppose when the question states that one gear is mounted directly over another, it actually means that they are in a mesh arrangement.

If they weren't then there is no way of imagining one can rotate clockwise and the other goes anti clockwise.

Having said that, I would like to ask you guys whether you think this question is a GMAT type question.

I was curious about this question only because of my engineering background.

mikemcgarry · Answer

Dear 12bhang, Actually, although I focus on the GMAT now, I have a Physics background. I believe this is a very poor question, probably designed by someone who was trying to write a challenging question but who really does not good physical/mechanical intuition. One could imagine a scenario in which the gears are coaxial but rotating in opposite directions. For example, suppose there were a central iron pole, and there were two cylindrical sleeves, well lubricated, around the rod, and each gear was mounted on one of those sleeves. The central pole simply provides alignment and support, not motion. The motion of the gears would have to be driven by something else, belts or cables or something. That is a complicated, but conceivable scenario in which the motion of the gears could be in opposite directions and entirely independent. I believe the question is trying to describe something like that, but " on the same axle " is indeed misleading. If a GMAT math question is well written, one does not need any external knowledge of engineering, economics, etc. etc. A good GMAT math question is a complete entity unto itself, unambiguously providing all the data you need to answer the question. This question falls abysmally short of that standard. Does all this make sense? Mike :-)

MrJglass · Answer

So what do we do? It looks like there is an unresolved debate over this one. 
Can someone else state their opinion or should the question be ignored till further notice?

Posted from my mobile device

Shobhit7 · Answer

According to the Q: 
Speed of rotation is same (linear speed)
Diameters of Large : Small are in ratio: 4:1 (96:24)
So, 4 rotations on small gear = 1 rotation on large gear.
Or, 1 rotation on small gear = 0.25 rotation on large gear.
Direction of rotation: opposite (small anti clockwise, large clockwise)

To calculate: distance traveled by a notch on large gear when notch of large gear meets notch on small gear for the second time.

Instance #1:
Small gear 1 rot = notch back at 12 o clock
Large gear= 0.25 rot = notch at 3 o o clock i.e. 0.25 on the circumstance

Instance #2:
Small gear Another 0.5 rot= notch at 6 o clock
Large gear = 0.25/2 rot = notch at 4.30 on clock i.e. 0.25 + 0.125= 0.375 on circumference
 
Meeting Point:
Using relative distance and speed concept

Relative Distance between two notches is 1/8 of circumference on large gear= 0.125

Angular speed is in ratio- small:large= 4:1
Relative speed: 5x

Time to meet: 0.125/5x = 0.025x

In this time, distance traveled by large gear  speed  x: 0.025

Position of large gear is= 96pi/(0.375+0.025) = 96pi/0.4= 38.4pi

Ans C

Posted from my mobile device

gracie · Answer

ratio of large to small gear diameters is 4:1
thus, ratio of large gear to small gear revolutions is 1:4
at 1st notch meeting, large gear has made 1/5 revolution clockwise while small gear has made 4/5 revolution counter-clockwise
at 2nd notch meeting, large gear has made 2/5 revolution clockwise while small gear has made 8/5 revolutions counter-clockwise
2/5*96⫪=38.4⫪
C

MrJglass · Answer

Okay, thanks guys!

Posted from my mobile device

Tanmaymandot1 · Answer

Since the notches start in the same position and move in opposite directions towards each other,
they will trace a circle together when they pass for the first time, having covered a joint total of
. When the notches meet for the second time, they will have traced two full circles together
for a total of .
Since the circumference of the large gear is 4 times greater than that of the small gear, the large
notch will cover only 1/4 of the number of degrees that the small notch does.

Since the circumference of the large gear is 4 times greater than that of the small gear, the large notch will cover only 1/4 of the number of degrees that the small notch does.

So the large notch will have covered 144* when the notches pass for the second time. Since the circumference of the large gear is 96 pi, we can set up the following proportion to solve for the linear distance covered by the large notch:

144/360 = x/96 pi

x= 38.4 pi

Rickooreo · Answer

Since the notches start in the same position and move in opposite directions towards each other, 
they will trace a circle together when they pass for the first time, having covered a joint total of 
. When the notches meet for the second time, they will have traced two full circles together 
for a total of .
Since the circumference of the large gear is 4 times greater than that of the small gear, the large 
notch will cover only 1/4 of the number of degrees that the small notch does.

SAHILBISHT0603 · Answer

very easy -600-650 level

ratio of diameter=big:small=4:1
ratio of speed=big:small=1:4

now the direction movement is opposite--->at meeting point a total of angle of 360 degree would have been covered by both(360=0degree ;similar to initial condition);
 since ratio of speed is 4:1 similar will be the ratio of angle subttended.
so from above
5x=360 
x=72
so we are required to find meeing distance of second time meeting point
similarly, next meeting at 72 degree again
now,
total degree covered by big gear =72(1st meet)+72(2nd meet)=144

now distance=144/360*pi*dia(big gear)=38.4pi
hence (C)

A circular gear with a diameter of 24 centimeters is mounted directly

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