A horizontal clothesline is tied between 2 poles, 20 meters apart.
When a mass of 5 kilograms is tied to the middle of the clothesline, it sags a distance of 4 meters.
What is the magnitude of the tension on the ends of the clothesline?
[ I am not understanding the setup of the problem. If you can help will be awesome. ]
Thank You
Just did it.
Of course, I can help you understand the problem setup.
In this scenario, imagine that there is a horizontal clothesline tied between two poles that are 20 meters apart. The clothesline sags in the middle when a mass of 5 kilograms is tied to it. The question asks for the magnitude of the tension on the ends of the clothesline.
To solve this problem, we can use the concept of equilibrium. In equilibrium, the net force acting on an object is zero. In this case, the clothesline is not moving, so it's in equilibrium.
To find the tension on the ends of the clothesline, we need to consider the forces acting on it.
1. Gravity: The mass of 5 kilograms applies a downward force due to gravity. The magnitude is given by the formula F = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Tension: The tension in the clothesline is the force that keeps it from collapsing. Since the clothesline is in equilibrium, the tension on both ends of the clothesline must be equal.
When the clothesline sags, it forms a right-angled triangle. The horizontal distance from the center point to one pole is half of the total length of the clothesline. In this case, the distance is 10 meters. The vertical distance from the center point to the lowest point of the sag is given as 4 meters.
Now, using the properties of right-angled triangles, we can find the length of the clothesline. The length is the hypotenuse of the triangle, and we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the squares of the other two sides.
Let's label the length of the clothesline as L. Using the Pythagorean theorem:
L^2 = (10 m)^2 + (4 m)^2
L^2 = 100 m^2 + 16 m^2
L^2 = 116 m^2
L ≈ √116 m
L ≈ 10.77 m
With the length of the clothesline found, we can now calculate the tension in the clothesline using the concept of equilibrium.
The vertical components of the tensions cancel each other out, as there is no net force in the vertical direction.
The horizontal components of the tensions add up to balance the horizontal component of gravity.
Since the clothesline is symmetrical, the two horizontal components of the tensions are equal. Therefore, we can say:
2 * Tension * cos(θ) = 0
Where θ is the angle between the clothesline and the vertical direction. In this case, cos(θ) = 4 m / 10.77 m.
From this equation, we can determine the magnitude of the tension:
2 * Tension * (4/10.77) = 5 kg * 9.8 m/s^2
Now, we can solve this equation to find the magnitude of the tension by rearranging and solving for Tension:
Tension = (5 kg * 9.8 m/s^2) / (2 * (4/10.77))
Simplifying this expression gives us the magnitude of the tension on the ends of the clothesline.