Angle between force and displacement to get minimum work?

Answer 1

The work done by a force is the dot product of the force and the displacement of the point of application of the force i.e.

component of force in the direction of displacement * displacement

or

component of displacement in the direction of force * force

let W be the work done. Then :

W = F.S

W = |F|*|S|*cos(θ) ---- equation 1

where F and S are force and displacement in vector form and |F|, |S| are their magnitudes respectively. cos(θ) is the angle between line of action of force vector and displacement vector.

It is clear from the equation 1 that work will be minimum when cos(θ) is minimum. It is known from trigonometry that minimum value of cos(θ) is -1 which is for angle 1800 or π radians. So work done will be minimum when angle between force and displacement vector is π radians or 1800 i.e. when point of application of force is displaced exactly opposite to the direction of application of force.

I appreciate the last attempt to answer the question, but I think it is wrong. The answer should be 90°, because although cos180° = -1, the negative sign in

- FScos180° only signifies that the work done has taken away energy from the system, or it signifies that the work is done in the opposite direction, implying that work is done. When the angle is 90°, on the other hand, work done = 0, ie, no work is done at all. Thus the answer should be 90°.

(The above answers define minimum work differently.

The first answers is if minimum work is negative work, therefore if you do work in the opposite direction of the displacement, the work you do is negative and is less than 0. Minimum work here is the work that is the lowest number.

The second answer defines minimum work as the magnitudeof the work done being minimized. Since magnitude is always positive, minimum work in this case would always be zero. Which one is correct depends on the situation you are dealing with.)

Answer 2

Here we can use the following work done formula

W = F.sCosθ

where F and s are force and displacement respectively and θ is the angle between them.

W will be maximum when θ = 0o

and will be minimum at θ = 180o

Answer 3

-- Negative work is the result of force and displacement in opposite directions.

-- Zero work is the result of either force perpendicular to displacement, or zero force,

or zero displacement.