A fire helicopter carries a 640 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 42.0° with respect to the vertical. Determine the force exerted by air resistance on the bucket.
Draw a free body diagram for the bucket.
The cable tension force is T
T cos 42 = bucket weight (which you know)
T sin 42 = air resistance force
Divide the second equation by the first. Remember that sin/cos = tan, for any angle.
tan 42 = (air resistance force)/weight
Solve for the unknown force
To determine the force exerted by air resistance on the bucket, we first need to resolve the force vectors acting on the bucket.
The weight of the bucket can be calculated using the equation:
Weight = mass × gravitational acceleration
Given that the mass of the bucket is 640 kg and the gravitational acceleration is approximately 9.8 m/s², we can calculate the weight:
Weight = 640 kg × 9.8 m/s² = 6272 N
Next, we need to resolve the tension in the cable. The vertical component of the tension is equal to the weight of the bucket, while the horizontal component will provide the force exerted by air resistance.
The vertical component of the tension can be found using the equation:
Vertical tension = tension × cos(angle)
Given that the tension in the cable is equal to the weight of the bucket (6272 N) and the angle with respect to the vertical is 42°, we can calculate the vertical component of the tension:
Vertical tension = 6272 N × cos(42°) ≈ 4543 N
Since the helicopter is traveling at a constant speed, the net force acting on the bucket must be zero. This means that the force exerted by air resistance, which is in the horizontal direction, must balance out the horizontal component of the tension.
Therefore, the force exerted by air resistance on the bucket is equal to the horizontal component of the tension:
Force air resistance = tension × sin(angle)
Force air resistance = 6272 N × sin(42°) ≈ 4309 N
Therefore, the force exerted by air resistance on the bucket is approximately 4309 N.
To determine the force exerted by air resistance on the bucket, we need to use the concepts of centripetal force and gravitational force.
First, let's calculate the tension in the cable. The total force in the vertical direction is the sum of the gravitational force and the vertical component of tension in the cable:
F_vertical = F_gravity + T_vertical
The gravitational force is given by:
F_gravity = m * g
where m is the mass of the bucket and g is the acceleration due to gravity (9.8 m/s^2).
In this case, the mass of the bucket is given as 640 kg. Therefore:
F_gravity = (640 kg) * (9.8 m/s^2)
Next, let's calculate the vertical component of the tension in the cable. Since the cable makes an angle of 42.0° with respect to the vertical, the vertical component of tension is given by:
T_vertical = T * cos(42.0°)
where T is the tension in the cable.
Now, we need to find the tension in the cable. The tension provides the centripetal force required to keep the bucket moving in a circular path. The centripetal force is given by:
F_centripetal = (m * v^2) / r
where v is the velocity of the bucket and r is the length of the cable.
In this case, the velocity of the bucket is given as 40.0 m/s, and the length of the cable is given as 20.0 m. Therefore:
F_centripetal = (640 kg * (40.0 m/s)^2) / 20.0 m
Finally, the tension in the cable can be calculated by equating the centripetal force to the horizontal component of the tension:
F_centripetal = T_horizontal
T_horizontal = T * sin(42.0°)
Now, using these equations, we can find the value of T (tension in the cable), and substitute it into the equation for the vertical component of the tension (T_vertical = T * cos(42.0°)), to calculate the force exerted by air resistance on the bucket.