A newly planted tree is supported against the wind by a rope tied to a stake in the ground, as shown in the figure below. The force of the wind, though distributed over the tree, is equivalent to a single force F, as shown in the figure. Find the tension in the rope. Assume that neither the tree's weight nor the force of the ground on the tree's roots produces any significant torque about point O. (Let F = 101 N, D = 2.30 m and d = 1.34 m.)
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Also, we cannot see the figure.
To find the tension in the rope, we need to analyze the forces acting on the tree. From the problem statement, we know that the wind force F is acting on a single point, and it is equivalent to a distributed force over the tree. The tree is supported by a rope, and there is a stake in the ground.
First, let's define the point where the rope is tied to the tree as point A and the stake in the ground as point O. We will take the torque about point O.
Since the problem assumes that neither the weight of the tree nor the force of the ground on the tree's roots produces any significant torque about point O, the only torque we need to consider is the torque due to the wind force.
To calculate the torque, we need to know the perpendicular distance between the line of action of the force F and point O. Let's call this distance h.
In the given figure, we can see that the distances D and d and the perpendicular distance h form a right triangle. We can use the Pythagorean theorem to find h.
h^2 = D^2 - d^2
h^2 = (2.30 m)^2 - (1.34 m)^2
h^2 = 5.29 m^2 - 1.79 m^2
h^2 = 3.50 m^2
h = sqrt(3.50 m^2)
h ≈ 1.87 m
Now that we have the perpendicular distance h, we can calculate the torque (τ) about point O using the following formula:
τ = r x F
where r is the position vector from point O to point A, and x denotes the cross product.
Since the line of action of the force F passes through point A, the position vector r is essentially the distance between point O and point A, which is given by d.
τ = d * F * sin(θ)
where θ is the angle between the position vector r and the force vector F. In this case, θ is 90° because the force F is perpendicular to the position vector d.
τ = d * F * sin(90°)
τ = d * F
Now, we know that the torque due to the wind force is equal to the tension in the rope multiplied by its distance from the stake, which can also be expressed as:
τ = T * d
where T is the tension in the rope.
Therefore, equating the two expressions for torque, we have:
T * d = d * F
T = F
So, the tension in the rope is equal to the force of the wind, which is given as F = 101 N.
Hence, the tension in the rope is 101 N.