A helicopter flies over the arctic ice pack at a constant altitude, towing an airborne 195-kg laser sensor which measures the thickness of the ice (see the drawing). The helicopter and the sensor move only in the horizontal direction and have a horizontal acceleration of magnitude 3.39 m/s2. Ignoring air resistance, find the tension in the cable towing the sensor.
110 N
To find the tension in the cable towing the sensor, we need to consider the forces acting on the system.
1. Weight force (mg):
The weight of the laser sensor can be calculated using the mass (m) and the acceleration due to gravity (g). The weight force is given by the equation Fw = mg.
2. Tension force (T):
The tension in the cable is the force that the helicopter exerts on the sensor to keep it in motion. This force acts horizontally and opposes the horizontal component of the weight force.
3. Horizontal component of weight force (mg sinθ):
Since the helicopter and sensor move only horizontally, the vertical component of the weight force does not affect the tension in the cable, so we only consider the horizontal component. This component can be calculated using the formula mg sinθ, where θ is the angle between the vertical direction and the cable.
Considering the given problem, the helicopter and the sensor move in the horizontal direction with an acceleration of 3.39 m/s^2. Since the acceleration is horizontal, the tension in the cable balances the horizontal component of the weight force.
Hence, we can write the following equation based on Newton's second law:
T - mg sinθ = ma
Since we are not given the angle θ, we need to find it using the information provided. In this case, we can assume that the helicopter is flying in a straight line directly above the sensor, making θ = 90 degrees. Therefore, sinθ = 1.
Simplifying the equation, we get:
T - mg = ma
Now we can substitute the known values into the equation. The mass of the sensor is given as 195 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. The acceleration of the helicopter is given as 3.39 m/s^2.
T - (195 kg)(9.8 m/s^2) = (195 kg)(3.39 m/s^2)
T - 1911 kg·m/s^2 = 662.1 kg·m/s^2
T = 2573.1 kg·m/s^2
Therefore, the tension in the cable towing the sensor is approximately 2573.1 N.