# Minimum force

## Homework Statement

In the first photo , the author stated that the minimum F2 is when the line of action is perpendicular to the line of action of FR, while in the second photo , the author gave that the shortest F2 is when line of action of F1 is perpendicular to F2 .
Which is the correct one ?
I have tried to analyse the questions , both are correct.
Is there any particular way of answering this type of question ?

## The Attempt at a Solution

#### Attachments

• 01.png
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• 02.png
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Dr. Courtney
Gold Member
2020 Award
The approach to most 2-D vector problems is similar:

1. Compute (or express) the x and y components of each force in terms of knowns and variables.

2. Add up x components of all relevant forces acting on the object and set equal to desired quantity.

3. Add up y components of all relevant forces acting on the object and set equal to desired quantity.

4. Solve equations for the desired unknowns.

This approach should work here.

CWatters
Homework Helper
Gold Member
+1 to that. Work the problem and you should get the right answer.

Intuitively... In problem one..... In order to get Fr along the x axis F2 has to counter the vertical component of F1. The best way to do that is with F2 vertical. If F2 was at any other angle it would have to be larger in order to have the required vertical component.

I can't follow problem two. I suggest you post the whole problem statement.

haruspex
Homework Helper
Gold Member
2020 Award
Is there any particular way of answering this type of question ?
In each question you have one force that is completely known, one of known direction, and a force to be minimised in magnitude. In each case, one of the forces is the resultant of the other two, and the force to be minimised is minimised by making it normal to the force of known direction. (See if you can show this is generally true.)
The difference between the problems given is that in the second case the fully known force is the resultant and in the first it is the force of known direction that is the resultant.

+1 to that. Work the problem and you should get the right answer.

Intuitively... In problem one..... In order to get Fr along the x axis F2 has to counter the vertical component of F1. The best way to do that is with F2 vertical. If F2 was at any other angle it would have to be larger in order to have the required vertical component.

I can't follow problem two. I suggest you post the whole problem statement.

F2 has to counter the vertical component of F1.
what di you mean by that ? sorry , my english is not so good

In each question you have one force that is completely known, one of known direction, and a force to be minimised in magnitude. In each case, one of the forces is the resultant of the other two, and the force to be minimised is minimised by making it normal to the force of known direction. (See if you can show this is generally true.)
The difference between the problems given is that in the second case the fully known force is the resultant and in the first it is the force of known direction that is the resultant.
what do you mean bu that ? in the first case , the FR is made perpendicular to F2 (2 forces are of unknown mangitude, but known direction ) , in the second case, the F1 is made perpendicular to F2 ..(2 forces are of unknown mangitude, but known direction )
in both case , we already know the direction of FR, F1 and F2 ... what do u mean by force of known direction ?

+1 to that. Work the problem and you should get the right answer.

Intuitively... In problem one..... In order to get Fr along the x axis F2 has to counter the vertical component of F1. The best way to do that is with F2 vertical. If F2 was at any other angle it would have to be larger in order to have the required vertical component.

I can't follow problem two. I suggest you post the whole problem statement.
of course i know that , but how to figure out the angle one by one ? it takes time

haruspex
Homework Helper
Gold Member
2020 Award
what do you mean bu that ? in the first case , the FR is made perpendicular to F2 (2 forces are of unknown mangitude, but known direction ) , in the second case, the F1 is made perpendicular to F2 ..(2 forces are of unknown mangitude, but known direction )
in both case , we already know the direction of FR, F1 and F2 ... what do u mean by force of known direction ?
In 1, you have a force of known magnitude and direction, F1, a force of known direction but unknown magnitude, FR, and a force of unknown direction (to be determined), and unknown magnitude, F2. You want to find the direction of F2 which minimise its magnitude.
In 2, you have a force of known magnitude and direction, FR, a force of known direction but unknown magnitude, F1, and a force of unknown direction (to be determined), and unknown magnitude, F2. You want to find the direction of F2 which minimise its magnitude.
In each case, the solution is to make F2 normal to the force of unknown magnitude. In 1 that's normal to FR, in 2 it's normal to F1.

werson tan
In 1, you have a force of known magnitude and direction, F1, a force of known direction but unknown magnitude, FR, and a force of unknown direction (to be determined), and unknown magnitude, F2. You want to find the direction of F2 which minimise its magnitude.
In 2, you have a force of known magnitude and direction, FR, a force of known direction but unknown magnitude, F1, and a force of unknown direction (to be determined), and unknown magnitude, F2. You want to find the direction of F2 which minimise its magnitude.
In each case, the solution is to make F2 normal to the force of unknown magnitude. In 1 that's normal to FR, in 2 it's normal to F1.
is there any explaination for this ? In each case, the solution is to make F2 normal to the force of unknown magnitude

haruspex
Homework Helper
Gold Member
2020 Award
is there any explaination for this ? In each case, the solution is to make F2 normal to the force of unknown magnitude
That's what I asked you in post #4.
Originally, you seemed to be bothered by an apparent inconsistency between the two cases. I have shown that there is a way to view the question generically which makes the answers consistent. Whether that consistency reflects an underlying truth is another matter.

CWatters
Homework Helper
Gold Member
F2 has to counter the vertical component of F1.
what di you mean by that ? sorry , my english is not so good

They want Fr to be on the x axis so the vertical component of Fr must be zero (Fry =O). Therefore...

F1y + F2y = 0

F2y = -F1y