A stone is tied to a string (length = 0.450 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 18.0% larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.
To determine the speed of the stone, we need to use the concepts of circular motion and centripetal force.
In the horizontal case, the tension in the string is balanced by the weight of the stone acting vertically downwards. Therefore, we can equate these two forces:
Tension = Weight of the stone
In the vertical case, the tension in the string is larger due to the additional force required to counteract the weight of the stone and provide the centripetal force. Therefore, we can write the equation:
Tension (vertical) = Weight of the stone + Centripetal force
To simplify the problem, we can assume the mass of the stone cancels out in both cases. Therefore, we don't need to know the mass of the stone to find the speed.
Now, let's denote the tension in the horizontal case as T_h and the tension in the vertical case as T_v. We are given that the maximum tension in the vertical case is 18.0% larger than the tension in the horizontal case:
T_v = T_h + 0.18T_h
T_v = 1.18T_h
Next, we need to relate the tensions to the speed of the stone. The tension in the string provides the centripetal force required to keep the stone in circular motion. The centripetal force can be expressed in terms of the mass of the stone (m) and the speed (v) as:
Centripetal force = m * v^2 / r
where r is the radius of the circular path, which is equal to the length of the string in this case.
Substituting the tension expressions into the centripetal force equation:
T_h = m * v^2 / r
T_v = m * v^2 / r + m * v^2 / r
Since the masses cancel out, we can rewrite the equations as:
T_h = v^2 / r
T_v = (1.18 * v^2 / r) + v^2 / r
Now, we can solve for the speed of the stone (v).
1.18 * v^2 / r + v^2 / r = (1.18 + 1) * v^2 / r = 2.18 * v^2 / r
T_v = 2.18 * v^2 / r
We know that T_v is 18.0% larger than T_h, so we can write:
1.18T_h = 2.18 * v^2 / r
Simplifying the equation:
T_h = v^2 / r = (1.18 / 2.18) * T_h
v^2 = (1.18 / 2.18) * T_h * r
Now, we can solve for v by taking the square root of both sides:
v = sqrt((1.18 / 2.18) * T_h * r)
Using the given values, T_h = T_v / 1.18 and r = 0.450 m:
v = sqrt((1.18 / 2.18) * (T_v / 1.18) * 0.450)
Now, substitute the 18.0% increase in tension (T_v = 1.18T_h):
v = sqrt((1.18 / 2.18) * (1.18T_h / 1.18) * 0.450)
Simplify the equation:
v = sqrt((1.18 / 2.18) * T_h * 0.450)
v = sqrt(0.64678899082 * T_h)
Hence, to determine the speed of the stone, you need to know the value of the tension in the horizontal case (T_h) and substitute it into the equation v = sqrt(0.64678899082 * T_h).