Question

Rhonda keeps a 2.0 -kg model airplane moving at constant speed in a horizontal circle at the end of a string of length \(1.0 \mathrm{m} .\) The tension in the string is \(18 \mathrm{N} .\) How much work does the string do on the plane during each revolution?

Short answer

Answer: The work done by the tension in the string on the model airplane during each revolution is 0.

Step-by-step solution

Determine the angle between the tension and the displacement

Since the airplane moves in a horizontal circular path at the end of the string, the force of tension acts in the radial direction, which is perpendicular to the direction of motion (tangential direction). Therefore, the angle between tension and displacement is 90 degrees.

Figure out the formula for work done

The work done by a force (W) can be calculated using the formula: \(W = F \cdot d \cdot cos(\theta)\) Where F is the applied force, d is the displacement, and \(\theta\) is the angle between the force and the displacement. In our case, F is the tension in the string, d is the circumference of the circle, and \(\theta\) is 90 degrees.

Calculate the circumference of the circle

We know that the length of the string is equal to the radius of the circular path (1.0 m). We can calculate the circumference of the circle using the formula: \(C = 2 \pi r\) Where C is the circumference and r is the radius. Using the given radius, we get: \(C = 2 \pi (1.0 \,\text{m}) = 2 \pi \,\text{m}\)

Calculate the work done by the string during each revolution

Now we'll use the formula for work done and plug in the tension (18 N), the displacement (2π m), and the angle (90 degrees): \(W = 18 \,\text{N} \cdot 2 \pi \,\text{m} \cdot cos(90°)\) Since \(cos(90°) = 0\), we have: \(W = 18 \,\text{N} \cdot 2 \pi \,\text{m} \cdot 0 = 0\)

Interpret the result

The work done by the tension in the string on the model airplane during each revolution is 0. This result makes sense because the tension is acting perpendicular to the direction of motion, and the work done by a force is maximum when the force and displacement are parallel, and zero when they are perpendicular.