to increase the frequency by 2 octave (s), to 4 f , by what factor would you have to increase the tension

16

Explanation:

Since frequency in a stretched string f = (n/2L)√T/μ and n, L and μ are constant, f ∝ √T  where T = tension in string

Now f₂/f₁ = √T₂/√T₁

Since f₁ = f (frequency at tension T₁) and f₂ = 4f (since its frequency increases by 2 octaves to 4f at tension T₂).

So,

f₂/f₁ = √T₂/√T₁

4f/f = √T₂/√T₁

√T₂/√T₁ = 4

squaring both sides, we have

(√T₂/√T₁)² = 4²

T₂/T₁ = 16

T₂ = 16T₁

So, the tension would have to increase by a factor of 16

close lines strong magnet less close weaker

the displacement produced by the body.  

the amount of force subjected to the body.  

the angle between the direction of force and displacement