A fire helicopter carries a 520-kg bucket of water at the end of a 20.0-m long cable. Flying back from a fire at a constant speed of 40.0 m/s, the cable makes an angle of 38.0° with respect to the vertical. Determine the force exerted by air resistance on the bucket.
my mans drwls doesn't know the purpose of this website
Draw a free body diagram. Note that three forces act on the bucket: weight, air resistance and rope tension. Because the velocity is constant, the three forces must balance.
A bit of trigonometry will tell you what the air resistance is.
I do not want to deprive you of the learning experience of doing it yourself.
To determine the force exerted by air resistance on the bucket, we need to first analyze the forces acting on the bucket.
1. Weight force (mg): This is the force exerted on the bucket due to gravity. It can be calculated using the formula: weight force = mass x gravity, where the mass (m) is 520 kg and the acceleration due to gravity (g) is approximately 9.8 m/s^2.
weight force = 520 kg x 9.8 m/s^2 = 5096 N
2. Tension force (T): This is the force exerted by the cable. Since the helicopter is flying at a constant speed, the tension force must be equal to the weight force. Therefore, T = 5096 N.
3. Air resistance force (F_res): This is the force exerted by air resistance on the bucket. It acts opposite to the motion of the bucket.
To find the horizontal component of the air resistance force (F_res,x) and the vertical component of the air resistance force (F_res,y), we can use the following equations:
F_res,x = T x cos(θ)
F_res,y = T x sin(θ)
where θ is the angle made by the cable with respect to the vertical (38.0°).
With these values, we can calculate the force exerted by air resistance on the bucket:
F_res = √(F_res,x^2 + F_res,y^2)
F_res,x = T x cos(θ) = 5096 N x cos(38.0°) = 5096 N x 0.7880 ≈ 4018 N
F_res,y = T x sin(θ) = 5096 N x sin(38.0°) = 5096 N x 0.6157 ≈ 3134 N
F_res = √(4018^2 + 3134^2) ≈ √(16145024 + 9809956) ≈ √25954980 ≈ 5093 N
Therefore, the force exerted by air resistance on the bucket is approximately 5093 N.
To determine the force exerted by air resistance on the bucket, we need to analyze the forces acting on the bucket.
We can start by breaking down the forces into vertical and horizontal components.
1. Vertical Forces:
- The weight of the bucket acts downward vertically and can be calculated using the equation: weight = mass * gravity, where g is the acceleration due to gravity (approximately 9.8 m/s²).
2. Horizontal Forces:
- The horizontal component of the tension in the cable provides the force necessary to keep the bucket moving at a constant speed and counteracts the horizontal air resistance.
- The vertical component of the tension in the cable counteracts the weight of the bucket.
Given:
- Mass of the bucket (m) = 520 kg
- Length of the cable (r) = 20.0 m
- Speed of the helicopter (v) = 40.0 m/s
- Angle with respect to the vertical (θ) = 38.0°
To find the force exerted by air resistance (F_air), we need to find the horizontal component of the tension in the cable (F_horizontal).
1. Calculate the vertical component of the tension:
F_vertical = m * g
F_vertical = 520 kg * 9.8 m/s²
2. Find the tension in the cable using the Pythagorean theorem:
Tension (T) = sqrt(F_vertical² + F_horizontal²)
3. Determine the horizontal component of the tension:
F_horizontal = T * cos(θ)
4. Calculate the vertical component of the force exerted by air resistance (F_air_vertical):
F_air_vertical = -F_vertical (opposite direction to weight)
5. Find the horizontal component of the force exerted by air resistance (F_air_horizontal):
F_air_horizontal = -F_horizontal (opposite direction to tension)
Finally, we can calculate the magnitude of the force exerted by air resistance using the Pythagorean theorem:
F_air = sqrt(F_air_vertical² + F_air_horizontal²)
By performing these calculations, we can determine the force exerted by air resistance on the bucket.