A helicopter flies over the arctic ice pack at a constant altitude, towing an airborne 142-kg laser sensor that measures the thickness of the ice (see the drawing). The helicopter and the sensor both move only in the horizontal direction and have a horizontal acceleration of magnitude 2.69 m/s2. Ignoring air resistance, find the tension in the cable towing the sensor.

Question

A helicopter flies over the arctic ice pack at a constant altitude, towing an airborne 142-kg laser sensor that measures the thickness of the ice (see the drawing). The helicopter and the sensor both move only in the horizontal direction and have a horizontal acceleration of magnitude 2.69 m/s2. Ignoring air resistance, find the tension in the cable towing the sensor.

Answer 1

To find the tension in the cable towing the sensor, we need to consider the forces acting on the system.

First, let's analyze the forces acting on the sensor:

1. Weight (mg): The weight of the sensor can be calculated by multiplying its mass (142 kg) by the acceleration due to gravity (9.8 m/s^2): w = (142 kg) * (9.8 m/s^2).

Next, let's analyze the forces acting on the helicopter:

1. Tension in the cable (T): This is the force exerted by the helicopter on the sensor through the cable. It is the force we want to find.

2. Thrust (T_helicopter): The helicopter exerts a forward thrust force to counteract the resistance and move forward. However, since the helicopter is moving with a constant velocity, we know that the thrust force is equal to the resistive force acting in the opposite direction. Therefore, T_helicopter = resistive force = mass * acceleration = (mass of the helicopter) * (acceleration).

Now, let's consider the horizontal acceleration of the system. Since the helicopter and the sensor are connected by a cable, their accelerations must be the same.

The given horizontal acceleration magnitude is 2.69 m/s^2. Since the sensor is airborne and not in direct contact with the ground, we can assume that the gravitational force on the sensor is balanced by the tension in the cable.

Therefore, we can equate the horizontal component of the tension force in the cable to the horizontal component of the weight of the sensor:

T * cos(θ) = w * sin(θ)

Here, θ is the angle between the tension force and the vertical direction. Since the sensor is flying at a constant altitude, we know that this angle is 90 degrees.

Now, let's substitute the expressions for T and w:

(T * cos(90°)) = (142 kg * 9.8 m/s^2 * sin(90°))

Simplifying:

T = 142 kg * 9.8 m/s^2

Therefore, the tension in the cable towing the sensor is 1391.6 N.