Reviewing Interactive Solution 5.46 will help in solving this problem. A stone is tied to a string (length = 1.29 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.
To solve this problem, we can follow these steps:
Step 1: Determine the tension in the string when the circle is horizontal.
We can use the formula for centripetal force:
F = m * a
where F is the tension in the string, m is the mass of the stone, and a is the centripetal acceleration.
In this case, the centripetal acceleration is given by:
a = v^2 / r
where v is the speed of the stone and r is the radius of the circle (which is equal to the length of the string in this case).
Since the stone is tied to the string, the tension in the string provides the centripetal force:
F = m * v^2 / r
Step 2: Determine the tension in the string when the circle is vertical.
In the vertical case, the tension in the string needs to support the weight of the stone as well as provide the centripetal force.
The weight of the stone is given by:
W = m * g
where W is the weight, m is the mass of the stone, and g is the acceleration due to gravity.
The tension in the string can be found by summing the vertical component of the tension (which balances the weight) with the horizontal component of the tension (which provides the centripetal force). Since the maximum tension in the string is 14.0% larger than the tension when the circle is horizontal, we can express this as:
T_vertical = (1 + 0.14) * T_horizontal
Step 3: Set up and solve the equations.
From Step 1, we have:
T_horizontal = m * v^2 / r
From Step 2, we have:
T_vertical = (1 + 0.14) * T_horizontal = (1 + 0.14) * (m * v^2 / r)
We can equate the two equations:
m * v^2 / r = (1 + 0.14) * (m * v^2 / r)
Simplifying, we get:
v^2 = (1 + 0.14) -> v = sqrt(1 + 0.14) -> v ≈ 1.12 (approximately)
Therefore, the speed of the stone is approximately 1.12 m/s.
To determine the speed of the stone, we need to use the information given in the problem and apply the concepts of circular motion.
Let's start by analyzing the horizontal case. In this scenario, the stone is whirled horizontally in a circle, and the tension in the string is denoted as T_horizontal. We are not given the value of T_horizontal, but we can find it based on other information.
The tension in the string can be related to the speed of the stone using the centripetal force formula:
T = (m * v^2) / r
where T is the tension, m is the mass of the stone, v is the velocity (speed) of the stone, and r is the radius of the circular path.
Since the stone is tied to a string, the tension in the string provides the centripetal force required to keep the stone in circular motion. Therefore, we can equate T_horizontal to the centripetal force.
On the horizontal circular path, the tension in the string is the only force acting on the stone in the vertical direction. So, T_horizontal is equal to the weight of the stone (mg), where g is the acceleration due to gravity (9.8 m/s^2).
T_horizontal = mg
Now, let's move on to the vertical case. In this scenario, the stone is whirled vertically in a circle. The maximum tension in the string is given as 14.0% larger than the tension in the horizontal case.
T_vertical = T_horizontal + 0.14 * T_horizontal
= T_horizontal * (1 + 0.14)
= T_horizontal * 1.14
We already know that T_horizontal = mg.
T_vertical = mg * 1.14
Since the stone is whirled in a circular motion, the tension in the string provides the centripetal force. Therefore, T_vertical can also be related to the velocity of the stone using the same centripetal force formula:
T_vertical = (m * v^2) / r
Now we can equate the expressions for T_vertical:
mg * 1.14 = (m * v^2) / r
The mass (m) cancels out, allowing us to solve for v:
1.14 * g = v^2 / r
To find the speed of the stone, we need the value of g (acceleration due to gravity) and r (the length of the string).
Substituting g = 9.8 m/s^2 and r = 1.29 m into the equation, we can calculate the speed of the stone:
1.14 * 9.8 = v^2 / 1.29
10.872 = v^2 / 1.29
v^2 = 10.872 * 1.29
v^2 = 14.0289
Taking the square root of both sides, we get:
v = √14.0289
v ≈ 3.748 m/s
Therefore, the speed of the stone is approximately 3.748 m/s.
You can also refer to Interactive Solution 5.46 for a visual representation and further explanation of this problem.