We are going to be investigating the forces on this toy fighter jet, which is strangely enough propeller driven. We are going to determine the speed of the jet in two different ways. In one method we will look at the forces acting on the airplane
and use Newton's Laws, and figure out the speed and acceleration that way. In the second method, we will measure the velocity directly by finding out how far the jet travels in a certain amount of time, and divide the distance by the time. And then we will
compare the two methods.
So one measurement that we are going to have to have to do this is the length of the string that the airplane is hanging on. So let's do that first before we put the plane back in motion. I will just put a meter stick up beside the string.
Start from the pivot point, and it almost comes down to the airplane. It takes another, we will estimate, probably a couple of centimeters to go to the plane. So that is 102 centimeters or 1.02 meters.
Now, let's take a look at the theory before we take the measurements. Let's take a look at the forces acting on the toy plane from the side. Here we have the weight of the airplane, we will call that mg, that is the mass of the plane times
the acceleration due to gravity, and the string pulls on it through a tension force. Now vertically, these are the only forces acting on the airplane. We want to combine these and do a net force analysis in order to determine the velocity of the airplane.
And to get its velocity we need the acceleration of the airplane. First of all, we know the airplane is moving in a horizontal circle and that when objects move in circles, there is acceleration is centripetal and directed toward the center of the circle.
So, the acceleration vector is like that. It makes sense, when doing a net force diagram, to make one of the axes point in the direction of acceleration. So, I'll make the x axis point that way, and the y axis just ahs to be perpendicular to that, it can point
up or down. So, the positive axes are in these directions.
Now, with those axes let's look at the components of Tension force. I am going to define the angle that the string makes with the vertical as angle θ, and that also makes that angle equal to theda because they are alternate interior angles.
That makes this side equal to Tsinθ, and this side Tcosθ. Now, we are ready to write our net force equation. In the X direction we only have one force, the horizontal component of Tension force. That's Tsinθ. Let's look at the vertical direction.
We have two vertical forces in the net force in the vertical direction. We have the vertical tension component, that's Tcosθ, and we have the full weight force, which is in the other direction, so its negative. Let's do some physics and some algebra to
combine these two equations.
First we know that the object is not accelerating vertically, it always stays in the same vertical plane, so f net y is equal to zero. And we know that f net x is equal to ma, by Newton's second law. So let's write down two equations.
ma = Tsinθ
mg = Tcosθ
We want to solve these to get rid of the Tension force, because we don't have a way to directly measure tension force. Well, its easy to do that if we divide one equation by the other.
ma/mg = Tsinθ/Tcosθ
a/g = tanθ
So the a is equal to gtanθ. That's a centripetal acceleration, so its possible to express *a* as v^{2}/r. And finally to solve for v, we get
v = sqrt(grtanθ)
So this is one method we will use to find the speed of the airplane. We will need to know the radius of the path (r) and the angle of the path (θ). What I measure before was the length of the string, that's the hypotenuse of this triangle.
In order to get the angle and the radius, and the radius is right here, we are going to measure this height. This is a right triangle right here, so if we know the height here, and the length here, we can use trig and the Pythagorean theorem to calculate the
angle of the path and the radius of the path respectively. So we can know all the numbers to calculate the speed of the airplane using this method.
Now for the other method, let's look at the top view of the airplane. It looks like a circle, and the radius of the path is "r" and the speed of the airplane is "v". Now, this particular method is particularly simple, as it involves figuring
out how far the plane travels in a certain amount of time. Now, we are going to use the formula that v = change in distance/change in time. Since this is a circular path, we can use a special formula for the change in distance, we can just use the circumference
of the path and the time it takes to go around once. From geometry, we know that circumference is 2πr. So this results in
v = 2πr/T ; where T is equal to the period of one circle.
So this will give us a second measurement for speed, which we can compare to Newton's Laws. These should come out about the same.
Here we are again with the airplane. We need to measure two things, the vertical height and the time it takes to go around. To get the vertical height, I'll take my meter stick and just put it up to the side here and bring it in as close to
the airplane as I can. And you can see here, about how high the plane is here on the meter stick. Zero is at the ceiling, so that tells you how low the plane is below its anchor point.
To get the time measurement, you can simply do that with your real player window, because you have a time display in your real player. So what you will do is watch the airplane and count the number of rotations, I would say 10 rotations. And
when you begin counting at zero, note the time in the real player and note the time at the 10^{th} rotation. You can take the total time and divide it by ten to get the period of the airplane.
So this gives you all the measurements you need to calculate the speed of the airplane using the two methods described. Its up to you now to make the calculations and the comparison of the two results. |