Conservation of Energy and Inertia resist acceleration

[zenite]

Hi Guys, I am doing an experiment to prove
1) Conservation of Energy
2) Inertia resist acceleration

The experiment is simple, rolling objects down a ramp.

So I timed the time taken for each object to reach the finishing line from rest.

To prove conservation of energy, I increased the height of the ramp, and then the steepness of the slope (height remains the same).

So when the slope steepness changes, the final velocity (hence kinetic energy) remains the same for an object. When height changes (steepness remains constant), the final velocity changes (potential energy changes, hence final kinetic). Is this reasonable to conclude that conservation of energy is true? The object is assumed to be pure rolling, hence no frictional effects.

And then for inertia. I used different shapes, solid sphere, hollow sphere and hollow cylinder. I got different timing for the shapes. The larger the inertia the longer the time taken. So how do I go about explaining inertia resist acceleration?

1 thing that confused me here is that how can I calculate the final velocity of the object with the data I had. I have height, degree of slope, displacement, mass. How do I get the final velocity? Can anyone help here?

My approach:
I used conservation of energy equation, PE1 = KE2 to find my final velocity. But can I really use it if I am trying to prove the theory is true? I then tried using kinematics to solve. But then I have to assume acceleration is constant. Is it constant? and what is the acceleration? Because inertia resist acceleration, each object have a different acceleration, correct? I thought a=g sin(theta) initially, but I am sure that's wrong.

 

[ZapperZ]

There is a problem.

You said that you are rolling objects down a ramp. The initial PE that you had at a particular height is being converted NOT just into the KE of the object via the translational motion, but also in the ROTATIONAL kinetic energy. So the energy is being converted into two different forms, not just into translational KE. To get the full conservation, you have to account for both.

Zz.

 

[zenite]

Yes, I do account for both. Will it affect the theory in any way?

The formula I used to calculate final velocity:

mgh = 0.5mv² + 0.5Kmv² where k is a constant dependable on shape of object (inertia)

so v = sqrt (2gh/(k+1))

Then I need to compare the theoretical velocity (the one above) with the practical one. problem is, how do I solve for final velocity? I have displacement and time. Is that sufficient? Can I can assume acceleration is constant, and if so what is the acceleration? Pls advice.

 

[ZapperZ]

Yes, I do account for both. Will it affect the theory in any way?

The formula I used to calculate final velocity:

mgh = 0.5mv² + 0.5Kmv² where k is a constant dependable on shape of object (inertia)

Where did you get that? You need to look up the expression for rotational KE. The fact that you are using the same "v" for both makes it incorrect.

Zz.

 

[zenite]

Sorry, I skipped some steps. Here is the full working, pls let me know if its wrong.

mgh = 0.5mv² + 0.5Iw²

For pure rolling, v = rw

mgh = 0.5mv² + 0.5(kmr²)(v/r)² = 0.5mv² + 0.5kmv²

The inertia formula is used as kmr², if its a sphere, k=1.


For the acceleration down a ramp for a rolling object, I did some research and found this:

Rolling objects have lower translational acceleration than that of a sliding, hence a is not equals to g*sin(theta). From a website, it states that a = sin(theta)*g*5/7

Can anyone tell me where does the 5/7 comes from? It seems like it comes from the moment of inertia, so the formula only applies to a certain shape. How do I derive the acceleration? Is the acceleration of a rolling ball even constant in the first place? I would really appreciate it if someone clears my doubt on the last question.