Feeling Weightless




When an object is in free fall, it can feel "weightless".

Feeling weightless is exactly that- feeling. You yourself are not actually losing or gaining mass. You just feel as though you are floating. You might be thinking,"Well, how does that work?" It is actually not that complicated. Let's take a look at this with twins Bob and Sara:


Screen_shot_2012-03-15_at_8.21.34_PM.png
Both Bob and Sara weigh (on earth) 60 kg.
Screen_shot_2012-03-15_at_8.19.43_PM.png
Here we see Bob swinging on a swing. At the top of his arc, he is "weightless" meaning he is in free fall, which means that the only force acting on him is Fg. For a moment, Bob is not moving up or down and has a Fn of zero.
Screen_shot_2012-03-15_at_8.19.58_PM.png
Above, we see Sara. She is jumping on a trampoline. When Sara reaches the top, there is a moment where she, too, stops and is not moving up or down. At the top of her jump, she also feels "weightless."

Apply it




Here is a problem we can apply this knowledge to:

The car (pictured below) is driving (in a positive direction) over a hill with a radius of curvature of 25 meters. At what constant speed would the car have to be driving for a 75 kg driver to feel weightless?

Screen_shot_2012-03-04_at_4.36.37_PM.png
Here, a car travels at a constant velocity (v) over a hill with a radius (r).

After reading the problem, we can not only use what we learned from Feeling Weightless, but we can also use The Expert Method to solve this problem.

Steps




1.Question: What is the problem asking?In this problem we are looking for constant speed.

2.Type: Dynamics- We are looking at the "How" part of physics.

3.Model: Constant Acceleration and Circular Motion- In this problem, as stated before, we will use equations dealing with velocity.

4.Pictorial Representation: Above, we have a picture given to us by the maker of the problem. Now, we can begin to add to our understanding of the problem by adding to the picture, listing our variables, and label the things we know.

Let's re-examine the picture from before. In the problem, it names the radius of the hill (r) as 25m long. When referring to a radius, we can think of the hill as being part of a circle, so lets add that to our picture... red.

Screen_shot_2012-03-04_at_4.59.09_PM.png
Now, we can begin to think of the hill as part of a circle and apply circular motion to our understanding of the problem.
Let's also make a list of known and unknown variables.So far we know the cells in the "Known Variables" section of the table below.

Known Variables
Unknown Variables
r = 25 meters

This is stated in the problem-
"a hill with a radius of curvature of 25 meters".
V = ?

We are trying to find the constant velocity at which the car needs to move.
m = 75 kilograms

This mass refers to the driver, also stated in the problem- "a 75 kg driver".

F(N) = 0

Normal Force on the driver by the car must equal zero in order for the driver to feel weightless.

F(C) = F(Net)

Centripetal Force is the same (equal to) Net Force. This is the definition of Centripetal Force.

F(C) = m*v^2
r

This equation (Centripetal Force = mass * velocity squared all divided by radius) we learned when discussing Centripetal Force.


5.Physical Representation: Now, we can draw a graph, motion map, and force diagram for the above problem to further analyze our understanding of the problem.

Graph:

Here, we can draw a graph showing the relationship between velocity and distance traveled.
Screen_shot_2012-03-04_at_5.35.56_PM.png

Because the car is traveling at a constant velocity, the graph has a straight line. A straight line on a graph, as referenced in Graphs, means the velocity is constant. This line's slope (m) is unknown because we have not yet found it, but we do know that the car is traveling in a positive direction.

Motion Map:
Screen_shot_2012-03-04_at_5.29.19_PM.png
As stated before, we know the car is moving in a positive direction with a constant, positive velocity. This is shown by congruency marks on the arrows.

Force Diagram:

First we might draw this:
on driver by car
Screen_shot_2012-03-04_at_5.44.45_PM.png on driver by earth

However, we have already decided that the F(N) or Normal Force is equal to zero, so it is not necessary to include it in our diagram. A better diagram would be this:

Screen_shot_2012-03-04_at_5.45.05_PM.pngon driver by earth


6.Mathematical Representation:

Finally, we can begin to solve the problem.

We have already decided that:

Screen_shot_2012-04-27_at_4.40.02_PM.png

We can first find the value of F(G) on the driver by (the equation) :

Screen_shot_2012-04-27_at_4.42.50_PM.png

We can also find F(Net) because the only unbalanced (or Net Force) is F(G) :

So,

Screen_shot_2012-04-27_at_4.45.23_PM.png

Since Screen_shot_2012-04-27_at_4.45.29_PM.png,

Screen_shot_2012-04-27_at_4.45.33_PM.png

We can now use Centripetal Force equation to find the velocity:

Screen_shot_2012-04-27_at_4.34.54_PM.png


7.Evaluation:

Lastly, we check for sign, magnitude, and unit.

Screen_shot_2012-04-27_at_5.01.53_PM.pngSIGN- The problem first states that the car is moving in a positive direction. Our velocity should also be positive.
Screen_shot_2012-04-27_at_5.01.53_PM.pngMAGNITUDE- Our answer (15.65 m/s) is similar to 35 mph. This is a reasonable speed for a car to be going over a hill.
Screen_shot_2012-04-27_at_5.01.53_PM.pngUNIT- The unit for velocity is m/s, which is what we used.