Elastic Strings

A particle of mass 3kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 14.7 N. The other end of the string is attached to a fixed point. The particle is held at equilibrium by a horizontal force of magnitude 9.8 N with the string inclined to the vertical at an angle θ.
c.) If the horizontal force is removed, find the magnitude of the least force that will keep the string inclined at the same angle.
Not sure in which direction the least force would act in. Help would be appreciated.

The angle of the string is fixed. You have three forces acting on the particle, it's weight being constant. You can change the magnitude of the string's tension, but not its direction. How do you minimise the force that you have to apply, to maintain equilibrium?

Hint: The three forces sum to zero, so form a triangle.You can change the length of the tension side. What angle does your applied force make with this to minimise its magnitude?

https://ibb.co/0Qtwf4p
I drew the diagram and varied the magnitude, but I don't see in which direction I should add the applied force.

What happens to the magnitude of the applied force as you vary its direction? Try moving further down.

(You've marked the arrow on it in the wrong direction)

I drew a free body force diagram for the tension and the weight. I haven't added the applied force. Here's the new diagram:

https://ibb.co/yY4FhV7
So would the applied force be in the direction indicated ?

No. As you move the applied force down the tension direction, what happens to its magnitude?

What do you mean by "down the tension direction"? Shouldn't the horizontal component of the applied force oppose the horizontal component of the tension? ( for equilibrium)

Yes, hence why the three forces form a triangle: T+W+F=0

Draw W and the direction of T. From the top of W, what direction gives the shortest distance to the direction of T?

Wait actually, I think the direction of the applied force is reversed on the diagram.

When you say "the top of W" does that mean from the arrow head of W?
Here is the new diagram:
https://ibb.co/JQy0HF0

Here's a diagram:


Note that T is showing the direction - its magnitude will vary to close the triangle (i.e. cause the net force on the particle to be zero). I've shown three possible vectors for F. We're wanting to minimise its magnitude, so, effectively looking for the least distance between its ends - the top of W and the direction of T. Hopefully it's now clear that this occurs when F and T are normal (i.e. at 90 degrees) to each other.