Are Moment Arm & Common Perpendicular Different?

[Diptangshu]

Not only the Problem I have attached in Picture, but there are few more where I get different answer while solving by taking Moment Component and by Common Perpendicular.

[NB : The problem Stated in 'Engineering Mechanics' by Ferdinand L. Singer. Process 3 is what I did, Process 1 is what is done in the Book & Process 2 is done for Verification by me]
It is clear that the Book is correct, but I cannot find out why my answer [Process 3] is not Matching.

No book can Clarify the Issue except a Real Teacher. I am Confused & Waiting eagerly for the Answer.

 

[haruspex]

I haven't found the error yet, but I see what the difference is between the two results. In process 1 there is effectively a division by |DA||CB|, where in process 3 the division is by |DAxCB|.

 

[Diptangshu]

I haven't found the error yet, but I see what the difference is between the two results. In process 1 there is effectively a division by |DA||CB|, where in process 3 the division is by |DAxCB|.

Yes, me too discovered the same ratio. And according to magnitude laws... Moment Arm is always shorter than Common Perpendicular.
But they should have been the Same! Very Intriguing.

 

[haruspex]

Moment Arm is always shorter than Common Perpendicular.

Doh! That's it.
If you view the system along the line of the perpendicular distance, the force does not appear quite orthogonal to BC. You have to restrict to the component that is.

 

[Diptangshu]

I haven't found the error yet, but I see what the difference is between the two results. In process 1 there is effectively a division by |DA||CB|, where in process 3 the division is by |DAxCB|.

Yes, me too discovered the same ratio. And according to magnitude laws... Moment Arm is always shorter than Common Perpendicular.
But they should have been the Same! Very Intriguing.

Doh! That's it.
If you view the system along the line of the perpendicular distance, the force does not appear quite orthogonal to BC. You have to restrict to the component that is.

I Think so... There is no other way to Explain it.
But it is quite hard to visualize in mind. Because common Perpendicular should have been shortest distance and hence it should be Moment arm,as in 2 Dimensional Plane. Why they don't match in 3 Dimensions... It's weird!

 

[haruspex]

common Perpendicular should have been shortest distance and hence it should be Moment arm

That certainly gives the moment about the nearest point on BC, and in a two dimensional plane that would be the answer. But here that moment would not be parallel to BC.
It might be clearer if you move B so that BC and AD are parallel. The perpendicular distance is still nonzero, but the moment about BC is zero.

 

[Diptangshu]

That certainly gives the moment about the nearest point on BC, and in a two dimensional plane that would be the answer. But here that moment would not be parallel to BC.
It might be clearer if you move B so that BC and AD are parallel. The perpendicular distance is still nonzero, but the moment about BC is zero.

Thank you.
This must be the explanation.

But this confusion is never encountered in general Engineering Mechanics course. Nowhere it is said, that the two lengths are different.
Very interesting fact is discovered here.

The factors |DA||BC| & |DA×BC|... Their physical difference is interesting.

Oops... for 0<Ѳ<90