A stone is tied to a string (length = 1.25 m) and whirled in a circle at the same constant speed in two different ways. First, the circle is horizontal and the string is nearly parallel to the ground. Next, the circle is vertical. In the vertical case the maximum tension in the string is 14.0% larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.

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To determine the speed of the stone, we can make use of the concept of centripetal force.

In the horizontal case, the tension in the string is the only force acting on the stone to keep it moving in a circle. Let's call this tension force "T". The centripetal force required to keep an object moving in a circle of radius "r" with speed "v" is given by the equation:

F = mv^2 / r

where m is the mass of the stone.

In the vertical case, apart from the tension force, an additional force is acting on the stone due to the weight of the stone. Let's call this force "W". The tension force must now not only provide the centripetal force but also counteract the weight of the stone. So, the maximum tension force, which occurs at the bottom of the circle, is equal to the sum of the centripetal force and the weight of the stone:

T_max = mv^2 / r + W

Given that the maximum tension in the vertical case is 14.0% larger than the tension in the horizontal case, we can write:

T_max_vertical = T_horizontal + 0.14 * T_horizontal

Simplifying further:

T_max_vertical = 1.14 * T_horizontal
mv^2 / r + W = 1.14 * T_horizontal

Since T = T_horizontal, we can rewrite the equation as:

mv^2 / r + mg = 1.14 * T

Now, consider the weight of the stone, which is equal to its mass multiplied by the gravitational acceleration "g":

W = mg

Substituting W in the equation:

mv^2 / r + mg = 1.14 * T

mv^2 / r + mv^2 / r = 1.14 * T

2 * mv^2 / r = 1.14 * T

Now, let's solve for the speed "v":

v^2 = (1.14 * T * r) / (2 * m)

v = sqrt((1.14 * T * r) / (2 * m))

Given that the length of the string is 1.25 m, and the increase in tension is 14.0% (or 0.14), we can plug in the known values to calculate the speed.