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.
26N
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To find the tension in the clothesline, we need to consider the forces acting on the pigeon.
Since the pigeon is in equilibrium, the sum of the vertical forces acting on it must be zero.
There are two vertical forces acting on the pigeon:
1. The weight of the pigeon acting downwards with a force of 1kg * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
2. The tension in the clothesline acting upwards.
We can express the tension in the clothesline as T.
Since the clothesline sags by 1 meter, the vertical distance between the pigeon and the midpoint of the clothesline is 0.5 meters.
Using the concept of moments, we can apply the condition for equilibrium:
Sum of clockwise moments = Sum of anticlockwise moments
Considering the clockwise moment due to the weight of the pigeon:
Clockwise moment = (1kg * g) * 0.5m
Considering the anticlockwise moment due to the tension:
Anticlockwise moment = T * 0.5m
Since the moments balance each other, we have:
(1kg * g) * 0.5m = T * 0.5m
Therefore, we can calculate the tension (T):
T = (1kg * g) * 0.5m / 0.5m
T = 1kg * g
Substituting the value of acceleration due to gravity, we get:
T ≈ 1kg * 9.8 m/s²
Thus, the tension in the clothesline is approximately 9.8 Newtons (N).
To find the tension in the clothesline, we need to consider the equilibrium of forces acting on the pigeon and the clothesline.
First, we'll consider the vertical equilibrium. Since the pigeon is at rest, the sum of the vertical forces acting on it must be zero.
Let's denote:
- Tension in the left segment of the clothesline as T1.
- Tension in the right segment of the clothesline as T2.
- Weight of the pigeon as W.
Since the pigeon is at the middle of the clothesline, the vertical components of T1 and T2 balance each other out. Therefore, T1 and T2 each have a vertical component of W/2.
Now, considering the vertical equilibrium:
T1 cosθ - T2 cosθ = W/2
(T1 - T2) cosθ = W/2
(T1 - T2) = (W/2) / cosθ ----- (Equation 1)
Next, let's consider the horizontal equilibrium.
There are no horizontal forces acting on the pigeon, so the horizontal components of T1 and T2 should also balance each other out. Therefore, T1 sinθ and T2 sinθ should be equal.
T1 sinθ = T2 sinθ ----- (Equation 2)
Now, to find the value of θ, we can use the sag (vertical displacement) of the clothesline. In this case, the sag is given as 1 meter, and the distance between the supports is 10 meters. The sag is a proportion of the horizontal distance, which means:
sag / distance between the supports = θ
1 / 10 = θ
θ = 0.1 radians
Now, we can substitute the value of θ in Equations 1 and 2 to solve for T1 and T2.
From Equation 2:
T1 sin(0.1) = T2 sin(0.1)
T1 = T2 ----- (Equation 3)
Substituting Equation 3 into Equation 1:
(T1 - T2) = (W/2) / cos(0.1)
0 = (W/2) / cos(0.1)
W/2 = 0
W = 0
Since the weight of the pigeon is zero, the tension in the clothesline is also zero.
Therefore, the tension in the clothesline is zero.