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?
To find the net work done on the plane, we first need to find the initial and final kinetic energies of the plane.
1. Initial kinetic energy (KEi):
The formula for kinetic energy is KE = 0.5 * mass * velocity^2.
Given that the speed of the plane is 13.6 m/s and the mass is 1.20 kg, we can calculate the initial kinetic energy.
KEi = 0.5 * 1.20 kg * (13.6 m/s)^2
2. Final kinetic energy (KEf):
After the plane speeds up, the tension in the guideline becomes four times greater. This means the plane maintains the same speed of 13.6 m/s but along a new circle with a smaller radius of 10.6 m.
Since the speed is the same, the final kinetic energy will also remain the same as the initial kinetic energy.
KEf = KEi = 0.5 * 1.20 kg * (13.6 m/s)^2
3. Net work done (W):
The net work done on an object is equal to the change in kinetic energy.
In this case, the change in kinetic energy is zero since the final kinetic energy (KEf) is the same as the initial kinetic energy (KEi).
Therefore, the net work done on the plane is zero.
W = 0
So, the net work done on the plane is zero.
To determine the net work done on the plane, we need to find the change in kinetic energy of the plane.
The formula for kinetic energy is given by:
Kinetic Energy = (1/2) * mass * velocity^2
Let's start by finding the initial kinetic energy of the plane when it is flying at a speed of 13.6 m/s on a horizontal circle of radius 19.1 m.
Initial Kinetic Energy (KEi) = (1/2) * mass * velocity_initial^2
= (1/2) * 1.20 kg * (13.6 m/s)^2
Next, let's find the final kinetic energy of the plane when the radius of the circle becomes 10.6 m and the tension in the guideline becomes four times greater.
Final Kinetic Energy (KEf) = (1/2) * mass * velocity_final^2 (after the radius change)
= (1/2) * 1.20 kg * velocity_final^2
Since the plane is moving in a circle, the centripetal force acting on it is provided by the tension in the guideline. Therefore, we can equate the centripetal force to the tension (T) in the guideline:
Centripetal Force = mass * acceleration
= mass * (velocity^2 / radius)
Tension (T) = mass * (velocity^2 / radius)
Given that the tension becomes four times greater when the radius changes, we can write:
4 * T_initial = T_final
Substituting the expressions for tension in both cases, we get:
4 * (mass * (velocity_initial^2 / radius_initial)) = mass * (velocity_final^2 / radius_final)
Canceling out mass and rearranging the equation, we get:
4 * (velocity_initial^2 / radius_initial) = velocity_final^2 / radius_final
From the above equation, we can solve for velocity_final. Once we have the final velocity, we can substitute it back in the equation for final kinetic energy to find KEf.
Once you have both initial and final kinetic energies (KEi and KEf), you can calculate the net work done on the plane using the formula:
Net Work Done = KEf - KEi