Calculating useful force or force at an angle

[richard9678]

Homework Statement

I don't understand my textbook.

Homework Equations

F_h = F_a Cos θ

Where F_a is some pulling force applied, in-between horizontal and vertical.

Where F_h is force in the horizontal.

The Attempt at a Solution

Intuitively, if the angle of the force applied is at 45°, then the force in the horizontal and vertical is the same, and half of F_a.

Cos θ for 45° is 0.707. I was expecting 0.5.

What am I getting wrong? Thanks.

 

[Doc Al]

Intuitively, if the angle of the force applied is at 45°, then the force in the horizontal and vertical is the same, and half of F_a.

You are correct that the horizontal and vertical components must be the same, but that doesn't mean that they are each half the total force. Realize that they are perpendicular. Imagine a 45°-45° right triangle with sides of 1 unit length. What's the hypotenuse equal?

 

[gneill]

Homework Statement

I don't understand my textbook.

Homework Equations

F_h = F_a Cos θ

Where F_a is some pulling force applied, in-between horizontal and vertical.

Where F_h is force in the horizontal.

The Attempt at a Solution

Intuitively, if the angle of the force applied is at 45°, then the force in the horizontal and vertical is the same, and half of F_a.

Cos θ for 45° is 0.707. I was expecting 0.5.

What am I getting wrong? Thanks.

Force components sum the way the legs of a right triangle "sum" to find the hypotenuse.

As you noted, if the angle is 45° then cos(θ) is 1/√2. But sin(θ) is also 1/√2, so the vertically directed force component is indeed equal to the horizontally directed component. But neither one is half of the net force!

For a right triangle with side lengths A and B, the hypotenuse C is given by the relationship:

C² = A² + B²

Force components add in the same fashion.

EDIT: Doc Al beat me to it!

 

[grzz]

... if the angle of the force applied is at 45°, then the force in the horizontal and vertical is the same, and half of F_a...

Yes, at that angle, the two components of the original force have the same magnitude.

BUT since these two components are not in the same direction the magnitude of each of them is not half the magnitude of the original force.

Two forces, each of magnitude F/2, can only give a resultant of magnitude F if these two forces are in the same direction.