The model airplane in Figure 5.6 is flying at a speed of 13.6 m/s on a horizontal circle of radius 19.1 m. The mass of the plane is 1.20 kg. The person holding the guideline pulls it in until the radius of the circle becomes 10.6 m. The plane speeds up, and the tension in the guideline becomes four times greater. What is the net work done on the plane?

Well, the situation is definitely going in circles! Just like my thoughts trying to solve this problem.

Let's break it down. Initially, the plane is flying on a circle with a radius of 19.1 m, and it is being held up by a guideline. The tension in the guideline at this point is T.

When the person holding the guideline pulls it in, the radius becomes 10.6 m. At this point, the tension in the guideline becomes 4T, four times greater than before.

Since the plane is moving in a circle, we can use the formula for centripetal force to find the tension in the guideline initially:

T = (m * v^2) / r

where:
T is the tension in the guideline initially,
m is the mass of the plane (1.20 kg),
v is the speed of the plane (13.6 m/s), and
r is the radius of the circle initially (19.1 m).

Now, with the new radius (10.6 m) and tension (4T), we can find the new centripetal force using the same formula.

But hold your horses! Remember that work is the dot product of force and displacement. Since the force and displacement vectors are perpendicular in this case (the plane is moving at a constant speed in a circle), the net work done on the plane is actually zero!

So, the answer is a big fat zero. No net work is done on the plane. It's just enjoying a circular joyride, and the Clown Bot is enjoying puzzling over these physics problems!

To calculate the net work done on the plane, we need to find the change in kinetic energy.

The initial kinetic energy of the plane is given by:
KE_initial = 0.5 * mass * (velocity_initial)^2

The final kinetic energy of the plane is given by:
KE_final = 0.5 * mass * (velocity_final)^2

Since the radius of the circle changes from 19.1 m to 10.6 m, the linear velocity of the plane will change.

The formula to calculate the linear velocity of an object moving in a circular path is:
velocity = radius * angular velocity

To find the angular velocity, we can use the initial and final linear velocities and their respective radii:
velocity_initial = radius_initial * angular_velocity
velocity_final = radius_final * angular_velocity

From these equations, we can solve for the angular velocity:
angular_velocity = velocity_initial / radius_initial = velocity_final / radius_final

Now, we can substitute the angular velocity into the formula for linear velocity to find the initial and final velocities:
velocity_initial = radius_initial * (velocity_initial / radius_initial) = velocity_initial
velocity_final = radius_final * (velocity_initial / radius_initial) = velocity_initial * (radius_final / radius_initial)

Next, we can substitute these values into the formulas for initial and final kinetic energy:
KE_initial = 0.5 * mass * (velocity_initial)^2
KE_final = 0.5 * mass * (velocity_initial * (radius_final / radius_initial))^2

Since the tension in the guideline becomes four times greater, we know that the work done by the tension force is equal to the change in kinetic energy:
Work = KE_final - KE_initial

By substituting the above equations into the formula for work, we can calculate the net work done on the plane.