A friend has some friends who own a tree farm in Foster. This
summer they cleared away all the trees that were within a 50m
distance from their house. This meant that their house was safer if
a fire started, as well as the fact that they now get more sun in winter – at the time the sun makes it through the clouds! There was one very big tree located between the house and the garage that was very tricky to fell. The tree was leaning at an 8° angle with respect to the vertical towards the garage. The center of mass of the tree was 1/3 of the way from the base of the tree. The entire tree was 66ft tall and weighed 4200kg. In order to prevent the tree falling on the garage, Steve climbed up the tree and attached a rope 6m of the ground. The other end of the rope was attached to a caterpillar on a hill resulting in the rope being parallel to the horizon. The caterpillar would start pulling as soon as Steve had made appropriate cuts thanks to his wife in the caterpillar patiently waiting for the signal to pull. This meant that the tree would fall the other way, and not on the garage. Steve did not want a stump and made the cut right on the ground.
a) How much tension would the caterpillar have to apply so that the tree falls away from the garage?
b) The safe working load for the rope is 8000N. Will the rope be safe to be used again (i.e. if the tension during the felling does not exceed the safe working load) after Steve has successfully felled the tree, or should he climb higher?
To answer these questions, we need to break down the problem into smaller parts. Let's start by finding the force required to pull the tree away from the garage.
a) To determine the tension that the caterpillar needs to apply to prevent the tree from falling on the garage, we need to consider the gravitational force acting on the tree. The angle of the tree's lean and the position of its center of mass are important factors in calculating this tension.
First, let's find the vertical component of the gravitational force acting on the tree. The weight of the tree, which we already know is 4200 kg, can be calculated using the equation W = m * g, where m is the mass of the tree and g is the acceleration due to gravity (approximately 9.8 m/s^2).
W = 4200 kg * 9.8 m/s^2
W = 41160 N
Since the tree is leaning at an 8° angle with respect to the vertical towards the garage, we can find the vertical component of the gravitational force by multiplying the weight by the sine of the angle.
Vertical component = W * sin(angle)
Vertical component = 41160 N * sin(8°)
Vertical component ≈ 7096.5 N
To find the tension that the caterpillar needs to apply, we need to balance this vertical component of the gravitational force. Since the rope is attached 6m above the ground, and the rope is parallel to the horizon, the tension in the rope will be equal to the vertical component of the gravitational force.
Therefore, the tension that the caterpillar needs to apply is approximately 7096.5 N.
b) We are given that the safe working load for the rope is 8000 N. If the tension during the felling does not exceed this safe working load, the rope can be safely used again.
Since the tension needed to prevent the tree from falling on the garage is 7096.5 N (as calculated in part a), which is less than the safe working load of 8000 N, the rope will be safe to use again. Steve does not need to climb higher.
Please note that these calculations assume ideal conditions and do not take into account any potential external factors or safety precautions. It is always important to prioritize safety and consult with professionals when dealing with potentially dangerous situations like tree felling.