By what factor would you have to decrease the tension in a stretched string in order to decrease the wave speed by a factor of 3.5?

I know that v = sqrt(T/u) but am confused as to what to do?

To reduce v by a factor of 3.5, T must decrease by a factor of 3.5^2.

The formula v = sqrt(T/u) tells you that.

To decrease the wave speed by a factor of 3.5, you need to decrease the tension in the string by a certain factor. To find this factor, we can rearrange the equation v = sqrt(T/u) to solve for T (tension).

1. Start with the equation v = sqrt(T/u).
2. Square both sides of the equation to eliminate the square root: v^2 = T/u.
3. Multiply both sides of the equation by u: v^2 * u = T.
4. Divide both sides of the equation by v^2: T = u * v^2.

Once we have the equation T = u * v^2, we can calculate the new tension (T') when the wave speed is decreased by a factor of 3.5.

1. Let the initial tension be T.
2. Let the initial wave speed be v.
3. Let the new tension be T'.
4. Let the new wave speed be v' = v/3.5.

Substituting these values into the equation T = u * v^2, we get:
T = u * (v/3.5)^2.

Now, we need to find the factor by which the tension must be decreased. This factor, denoted as F, is defined as:
F = T'/T.

Substituting the expressions for T and T' into the above equation, we get:
F = (u * (v/3.5)^2) / (u * v^2).

Simplifying, we get:
F = (1/3.5)^2.

Evaluating this expression, we find:
F ≈ 0.082.

Therefore, to decrease the wave speed by a factor of 3.5, you would need to decrease the tension in the string by a factor of approximately 0.082.