A stone is tied to a string (length = 0.600 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 20.0% 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 can use the principles of circular motion, namely centripetal force and tension.
Let's consider the horizontal case first. In this case, the tension in the string is equal to the centripetal force required to keep the stone moving in a circular path.
The centripetal force is given by the equation:
Fc = (m * v^2) / r
Where Fc is the centripetal force, m is the mass of the stone, v is the speed of the stone, and r is the radius of the circle.
In the horizontal case, the tension in the string is equal to the centripetal force:
T1 = Fc
Now let's consider the vertical case. In this case, the tension in the string is larger by 20% than the tension in the horizontal case:
T2 = 1.2 * T1
The centripetal force in the vertical case is still given by the same equation:
Fc = (m * v^2) / r
And the tension in the string is equal to the centripetal force:
T2 = Fc
Substituting the expression for centripetal force in terms of tension, we can equate the two equations:
1.2 * T1 = (m * v^2) / r
Now we can solve for the speed of the stone, v.
1.2 * T1 * r = m * v^2
v^2 = (1.2 * T1 * r) / m
v = sqrt((1.2 * T1 * r) / m)
To find the speed of the stone, we need values for the tension T1, the radius r, and the mass of the stone m. Once we have those values, we can substitute them into the equation and calculate the speed of the stone.