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.
I know how to figure it out. Just how do you get the radius?
isn't the radius the length of the string?
The speed of the stone when? It changes in the vertical circle.
To determine the radius of the circle, you can use the given length of the string and the difference in tensions between the horizontal and vertical cases.
Let's denote the tension in the horizontal case as T1 and the tension in the vertical case as T2. We are given that T2 is 18.0% larger than T1.
In the horizontal case, the tension in the string is equal to the centripetal force acting on the stone:
T1 = m * v1^2 / r ... (equation 1)
where m is the mass of the stone, v1 is the speed of the stone, and r is the radius of the circle.
In the vertical case, the tension in the string is equal to the sum of the centripetal force and the force due to gravity:
T2 = m * v2^2 / r + m * g ... (equation 2)
where v2 is the speed of the stone and g is the acceleration due to gravity.
Given that T2 is 18.0% larger than T1, we can write:
T2 = T1 + 0.18 * T1
T2 = 1.18 * T1 ... (equation 3)
Now, we can solve equations 1, 2, and 3 to find the value of r:
From equation 1: T1 = m * v1^2 / r
From equation 2: T2 = m * v2^2 / r + m * g
Substituting equation 3 into equation 2, we get:
1.18 * T1 = m * v2^2 / r + m * g
Simplifying and rearranging, we have:
(1.18 * T1) - T1 = m * v2^2 / r + m * g - T1
0.18 * T1 = m * v2^2 / r + m * g - T1
Substituting the expression for T1 from equation 1, we get:
0.18 * (m * v1^2 / r) = m * v2^2 / r + m * g - (m * v1^2 / r)
0.18 * m * v1^2 / r = m * v2^2 / r + m * g - m * v1^2 / r
Canceling out mass and rearranging, we have:
0.18 * v1^2 - v1^2 = v2^2 - v1^2 + g
Simplifying further:
0.18 * v1^2 - v1^2 - v2^2 + v1^2 = g
0.18 * v1^2 - v2^2 = g ... (equation 4)
Now, we can use equation 4 to find the value of v2, the speed of the stone. Rearranging equation 4, we have:
v2^2 = 0.18 * v1^2 - g
Taking the square root of both sides, we get:
v2 = sqrt(0.18 * v1^2 - g)
This equation gives you the speed of the stone when the circle is vertical.
To determine the speed of the stone, we first need to find the radius of the circle. In this case, the length of the string is given as 0.450 m.
When the stone is whirled in a horizontal circle, the tension in the string provides the centripetal force required to keep the stone moving in a circular path. The tension force in the string can be calculated using the equation:
Tension = (mass of the stone) × (centripetal acceleration)
In this case, the string is nearly parallel to the ground, so the angle between the string and the ground is small. As a result, the tension in the string can be approximated as equal to the weight of the stone:
T_horizontal = (mass of the stone) × (acceleration due to gravity)
Next, let's consider the vertical case. When the stone is whirled in a vertical circle, there are two forces acting on the stone: the tension in the string and the weight of the stone. At the bottom of the circular path, these two forces add up to provide the net inward force required for circular motion. The maximum tension in the string occurs at the bottom of the circular path, where the stone is experiencing its maximum acceleration.
Using the same equation as before, the tension force in the string at the bottom of the circular path can be calculated as:
T_vertical = (mass of the stone) × (centripetal acceleration) + (mass of the stone) × (acceleration due to gravity)
We are given in the problem that the maximum tension in the vertical case is 18.0% larger than the tension in the horizontal case:
T_vertical = T_horizontal + 0.18 × T_horizontal
Simplifying the equation:
T_vertical = 1.18 × T_horizontal
Substituting the expressions for the tensions:
(mass of the stone) × (centripetal acceleration) + (mass of the stone) × (acceleration due to gravity) = 1.18 × (mass of the stone) × (acceleration due to gravity)
Canceling out the mass of the stone:
(centripetal acceleration) + (acceleration due to gravity) = 1.18 × (acceleration due to gravity)
Since the acceleration due to gravity is the same in both cases, we can cancel that out as well. We are left with:
(centripetal acceleration) = 0.18 × (acceleration due to gravity)
The centripetal acceleration can be expressed in terms of the speed and the radius of the circle as:
centripetal acceleration = (speed)^2 / radius
Substituting this equation into the previous one, we have:
(speed)^2 / radius = 0.18 × (acceleration due to gravity)
Rearranging the equation, we can solve for the radius:
radius = (speed)^2 / (0.18 × (acceleration due to gravity))
Now, we can solve for the speed of the stone by plugging in the values for the radius, the acceleration due to gravity (approximately 9.8 m/s^2), and the tension in the horizontal case.