A fire helicopter carries a 555 kg empty water bucket at the end of a cable 20.9 m long. As the aircraft flies back from a fire at a constant speed of 38.4 m/s, the cable makes an angle of 38.3° with respect to the vertical.

A) After filling the bucket with sea water, the pilot returns to the fire at the same speed with the bucket now making an angle of 8.55° with the vertical. What is the mass of the water in the bucket?

To find the mass of the water in the bucket, we first need to determine the tension in the cable when the bucket is filled with water.

In this situation, we have a helicopter carrying the water bucket at a constant speed, so the net force in the vertical direction must be zero (assuming no vertical acceleration).

Let's break down the forces acting on the bucket when it is filled with water:

1. Gravity (weight): The weight of the bucket filled with water can be calculated using the formula W = mg, where W is the weight, m is the mass, and g is the acceleration due to gravity. The weight acts in the downward direction.

2. Tension in the cable: The tension in the cable acts in the upward direction.

These two forces together determine the angle of the cable with respect to the vertical.

Now, let's analyze the situation and find the tension in the cable when the bucket is filled with water.

Using trigonometry, we can determine that the vertical component of the tension force is given by T * sin(θ), where T is the tension in the cable and θ is the angle between the cable and the vertical.

The weight of the bucket with water is 555 kg * 9.8 m/s² (acceleration due to gravity), which gives us 5441.4 N (downward force).

Since the net force in the vertical direction is zero (the helicopter is flying at a constant speed), the upward force (T * sin(θ)) provided by the tension in the cable must balance the weight of the bucket:

T * sin(θ) = 5441.4 N

We now have an equation with two unknowns: T (tension) and θ (angle). However, we have another piece of information given in the problem: when the bucket is filled with water, the cable makes an angle of 8.55° with the vertical.

Substituting this value into the equation, we have:

T * sin(8.55°) = 5441.4 N

This equation allows us to solve for T, the tension in the cable.

Once we have determined the tension in the cable, we can calculate the force exerted on the water in the bucket, which is equal to the tension force.

Finally, we can find the mass of the water using Newton's second law of motion: F = ma. The force exerted on the water is equal to the mass of the water multiplied by the acceleration due to gravity (9.8 m/s²).

So, the mass of the water in the bucket can be calculated by dividing the force exerted on the water (tension force) by the acceleration due to gravity.

M = F / g

where M is the mass of the water, F is the force exerted on the water, and g is the acceleration due to gravity.

I hope this explanation helps you understand the process of finding the mass of water in the bucket.

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