A stone is tied to a string (length = 0.945 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 6.60% larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.
vertical:
Ftop = mg + Tt = m v^2/R
Fbot = Tb-mg = m v^2/R
so max is
Tb = m g + mv^2/R
so
mg +Tt = Tb-mg
2 mg = Tb-Tt LOL, could have guessed
Horizontal
T = m v^2/R
given
Tb = 1.066 T
mg + mv^2/R = 1.066 mv^2/R
g = .066 v^2/R
so
v^2 = g R /.066
To determine the speed of the stone, we need to consider the forces acting on it in both cases.
In the horizontal case, the only force acting on the stone is the tension in the string. Since it is nearly parallel to the ground, the tension is directed towards the center of the circle. We'll denote this tension as T_h and the speed of the stone as v_h.
In the vertical case, the tension in the string has two components: one directed towards the center of the circle and one directed upwards, opposing the force of gravity. Denoting the tension directed towards the center of the circle as T_v and the tension opposing gravity as T_g, we can write:
T_v = T_h + T_g
Since we know that the maximum tension in the vertical case is 6.60% larger than the tension in the horizontal case, we can write:
T_v = T_h + 0.066 * T_h = 1.066 * T_h
Now, let's consider the forces in the vertical case. The vertical component of the tension (T_g) is equal to the weight of the stone:
T_g = m * g
where m is the mass of the stone and g is the acceleration due to gravity. The weight of an object can be calculated using the formula:
Weight = mass * acceleration due to gravity
So,
m * g = Weight = m * g
Now, let's substitute T_g in T_v equation:
T_v = T_h + m * g
Since the tension in the string provides the necessary centripetal force for circular motion, we can equate T_v to the centripetal force:
T_v = m * v_v^2 / r
where v_v is the speed of the stone in the vertical case and r is the radius of the circle.
Similarly, we can equate T_h to the centripetal force in the horizontal case:
T_h = m * v_h^2 / r
Now, let's substitute these values and equations into the previous expression for T_v:
m * v_v^2 / r = T_h + m * g
m * v_v^2 / r = m * v_h^2 / r + m * g
v_v^2 = v_h^2 + g
From this equation, we can see that the speed of the stone in the vertical case is equal to the speed in the horizontal case plus the square root of the acceleration due to gravity.
To determine the speed of the stone, we need the values of g and v_h. The acceleration due to gravity is approximately 9.8 m/s^2, which is typically denoted as g. However, we don't have the value of v_h. Therefore, we need additional information or measurements to calculate the speed of the stone.