Please help find the solution to the following problem: A helicopter flies over the arctic ice pack at a constant altitude, towing an airborne 129 kg laser sens

Question

A helicopter flies over the arctic ice pack at a constant altitude, towing an airborne 129 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 2.84 m/s2. Ignoring air resistance, find the tension in the cable towing the sensor.

Units: N

Answer 1

What does Newton's second law say?

 

It takes a certain amount of net force to accelerate any mass, because according to the first law, objects with mass have an inertia that prefers to stay at rest if at rest and keep traveling at a constant velocity (in a straight line) if in motion.    In fact, the amount of net force necessary to accelerate an object (in other words, to change its velocity, by either changing its direction or its speed) must be equal to the product of the mass of the object and the time rate of change of the velocity.

 

The problem is asking you to find the tension in the cable that is pulling the sensor. Tensions in ropes and cables can only ever act in a direction along those items, so that pull is a force on the sensor in the direction of the helicopter. Another force on the sensor is its weight, the pull on the sensor due to the Earth's gravity.  Since the problem states that we can neglect air resistance, we have now listed all of the non-negligible forces on the sensor.

 

There is no way to post the picture but I assume it shows the sensor hanging below the helicopter at a certain angle (which we could also calculate).

 

You don't know the tension, but nonetheless you can assign it a value: T.  With a diagram showing the direction of the tension (towards the helicopter), the direction of gravity (down) and the direction of the acceleration (horizontal), you know the net force on the sensor is in the same direction as the acceleration with magnitude equal to the mass of the sensor * the magnitude of the acceleration.

Since two of these three vectors are horizontal and vertical, respectively, and thus already at ninety degrees relative to each other, while the third (the tension) is at some other angle that is perpendicular to neither of these, it makes the most sense to choose your coordinate system for the problem to be aligned with the horizontal and vertical vectors, because that way we need to only break one vector down into components aligned with our chosen coordinate system: the (unknown) tension.  I assume you know SOH-CAH-TOA and how to decompose a vector into its components ... if not please go back and learn that before attempting to solve more problems like this.

 

Once the tension is broken into components aligned with our coordinate system, we can consider the original tension vector to have been replace by those two components. So we now have four vectors: the horizontal component of the tension, the vertical component of the tension, gravity, and the net force.

 

The single horizontal component of the tension, which should be written in terms of T and a trig function (which trig function, sine or cosine, depends on where you chose to label an angle "theta" defining the orientation of the tow cable), must be equal to the net force since the vertical component of the tension must cancel the gravitational force.  We know this because if the vertical forces did not balance, the sensor would not be accelerating horizontally.

 

Therefore, we can set the horizontal component of the tension equal to the net force, and solve for T.

 

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