A 1kg pigeon sits on the middle of a clothesline whose supports are 10m apart. The clothesline sags by 1 meter. If the weight of the clothesline is negligible, find the tension in it.
To find the tension in the clothesline, we can use the principle of equilibrium. In equilibrium, the sum of the forces acting on an object is zero.
In this case, the tension in the clothesline acts upwards, and the weight of the pigeon acts downwards. The tension can be resolved into horizontal and vertical components.
Let's consider the vertical equilibrium first. The vertical component of the tension should balance the weight of the pigeon. The weight of the pigeon can be calculated using the equation:
Weight = mass × acceleration due to gravity
Weight = 1kg × 9.8m/s² = 9.8N
The vertical component of the tension in the clothesline should be equal to the weight of the pigeon:
Tension (vertical) = Weight of pigeon = 9.8N
Now, let's consider the horizontal equilibrium. Since the clothesline sags by 1 meter, a right-angled triangle is formed between the midpoint of the clothesline and the supports. The horizontal component of the tension in the clothesline should balance the horizontal component of the tension from both sides of the sag.
To find the horizontal component of the tension, we can use the Pythagorean theorem:
(horizontal component of tension)² = (total tension)² - (vertical component of tension)²
(horizontal component of tension)² = T² - (9.8N)²
(horizontal component of tension)² = T² - 96.04N²
Since the total length of the clothesline is 10m, the horizontal component of tension acts on 5m on each side of the sag. Therefore, the horizontal component of tension from each side of the sag is half of the total horizontal component.
(horizontal component of tension from each side) = √(T² - 96.04N²) / 2
Now, using the principle of equilibrium, we can sum up the horizontal component of the tension from both sides and set it equal to zero:
(horizontal component of tension from each side) + (horizontal component of tension from each side) = 0
√(T² - 96.04N²) / 2 + √(T² - 96.04N²) / 2 = 0
√(T² - 96.04N²) = 0
Simplifying the equation, we get:
T² - 96.04N² = 0
T² = 96.04N²
T = √96.04N²
T = 9.8N
Therefore, the tension in the clothesline is 9.8 Newtons.