Calculating Gravity from Inclined plane

1. May 23, 2015

Fat_Squirrel

1. The problem statement, all variables and given/known data
http://i.imgur.com/Osn6pKO.png

2m long Inclined plane. Height at top of the ramp is 6.1cm
We are given distances of 0.5, 1, 1.5 2.0m and their times (close to 2.5s, 3.5s ,4.25, 5s).

Need to calculate g.

2. Relevant equations
d=1/2at^2
g=a/(sinθ/(1+I/mr^2)) (where sinθ=0.061/2)

3. The attempt at a solution
So I plotted all the data in excel. Graphed time squared against distance and excel tells me the gradient of that slope is 0.0823, so a is twice that, 0.1646.

If I use g=a/sin(theta), 0.1646/(0.061/2) = 5.39 .. pretty far off 9.8.

So I assume I have to use the additional info they give me (the diameter of the ball and the V shaped ramp) and add in the effect of the ball rolling. But i'm not sure what formua to use. I thought I/mr^2 for a sphere is 2/5, but then my answer is even further from what I know to be g.

Suggestions?

2. May 23, 2015

Fat_Squirrel

Actually my answer gets closer, but i'm looking at g=7.58 ?? Seems a fair way off still to me. What am I doing wrong?

3. May 23, 2015

haruspex

It is not the same as a ball rolling down a flat plank. Please post the algebraic details of your calculation (don't plug in numbers).

4. May 23, 2015

Fat_Squirrel

I was going with a=5/7*g*sin θ, so g = 7a / 5*sinθ

How do I account for it not being equivalent to a flat plane ?

5. May 23, 2015

Fat_Squirrel

Or are you asking how I got the answer for a ? If so, I took average of the times, squared these times and plotted it against distance.
Excel then gave me line of best fit with gradient = 0.0823

As d=1/2at^2, we need to double the gradient to get a, so 0.1646.

6. May 23, 2015

haruspex

One way of arriving at the 5/7, for a flat plank, is to consider the rolling ball as rotating about its point of contact with the plank. Applying the parallel axis theorem, $mr^2+\frac 25 mr^2=\frac 75 mr^2$.
Where is the instantaneous axis of rotation in the present case? What does the parallel axis theorem give for that?

7. May 23, 2015

Fat_Squirrel

Honestly, no idea. Purely guessing I'd say its the midpoint on the chord between the two points of contact which I calculated to be very nearly 10.41mm from the centre of the ball.

Even if that's correct, I have even less of an idea what I would do with this figure.

8. May 23, 2015

Fat_Squirrel

1 + 10.4/15 = 1.69333 if I multiply that by my original a/sinθ , I get 9.14. My closest answer yet. But based on nothing but fact I know somewhat what I need to multiple this answer by to get close to g.

Tried watching some youtube videos, sadly they break into Indian at a moment's notice :(

9. May 23, 2015

dean barry

starting from scratch:
find the effective rolling radius of the ball in the groove.
(the distance between the ball centre and the centre of the line between the contact points)
Imagine the ball moving at a constant (arbitrary) 10m/s in the groove, and
calculate the linear KE and the rotating KE at this speed.
Then:
Calculate the effective mass (em) of the ball from:
em = ( 1 + ( rotating KE / linear KE ) * actual mass of the ball
The acceleration of the ball (a) should be :
a = f / em
(you know a from your data)
(f is the driving force due to gravity = m * g * sine ( incline angle)
jumble to isolate g

10. May 23, 2015

Fat_Squirrel

We aren't given the mass of the ball anywhere sadly. I must have read the question a half dozen times to make sure I didn't miss it.

11. May 23, 2015

dean barry

i set this up on excel with a fixed rolling radius and variable masses (solid sphere assumed in all cases) and the acceleration varies with the mass
i cant see how to solve it.

12. May 23, 2015

haruspex

If you get your parentheses right you will discover mass cancels out.

13. May 23, 2015

haruspex

Yes, the axis is the chord joining the points of contact.
But this doesn't just change the effective inertia. It also changes the relationship between the linear velocity and the angular velocity.
Suppose the chord is distance x from the ball's centre (please avoid plugging in numbers for now). What is the relationship between v and $\omega$? What is the moment about the axis? What does that give you for the acceleration?

14. May 24, 2015

Fat_Squirrel

Avoid plugging numbers in? I dont even know what formula i'm supposed to be using. I have no idea what you're talking about. Sorry, but I give up.
I'm going to need something beyond these cryptic clues. This is my first month of physics and this is all well above my head.

15. May 24, 2015

haruspex

As moment of inertia problems go, this is somewhat advanced.
Draw yourself a side view. Say the ball has radius R, and the centre of the ball is distance A from the chord joining the points of contact with the ramp. If at some instant the ball is rotating at rate $\omega$, what will be its linear speed down the ramp? What will its total KE be (linear plus rotational)?

16. May 24, 2015

Fat_Squirrel

Appreciate the assistance. But I've already spent WAY too long on this problem. It's only worth a single bonus mark. It's not worth my mental heath to continue it further. I'll just wait for the answer and live without that mark.

Thanks for the help though.

17. May 25, 2015

Fat_Squirrel

v= rω ??
Moment about the axis? No idea. I_com + Mr^2 ? Honestly, what am I missing?

Can you maybe explain this like I'm 5? I can't find anything discussing these ideas that dont include mass, and I can't see how to remove the mass from the equation, and I can't see how we calculate the inertia at a point off the centre without using mass I dont have.

How do we use the parrallel axis theorum withhout mass? i dont see how we can remove it from what Dean Barry suggested. And everything that derives these equations always uses mass. Textbook, youtube vids, physics forums aren't helping.

Can you please give me a big nudge in the right direction?

18. May 25, 2015

haruspex

No. Did you draw yourself a diagram as I suggested? You should have a circle radius r representing the ball, and a sloped line that cuts through it representing the line of contact with the ramp. At its closest point, the ramp is distance x from the centre of the ball. It might help to draw another circle concentric with the first, radius x.
As the ball rolls down the ramp, if it turns an angle theta, how far has it moved down the ramp?
Right general formula. In that formula, r represents the distance from the mass centre to the instantaneous centre of rotation. In the present case, the instantaneous centre of rotation is the chord, so its distance x from the mass centre.
In terms of m, r and x, what is I_com?
Just put the mass as an unknown m. You'll see what happens later.

19. May 25, 2015

Fat_Squirrel

I_com=2/5mr^2 ?
So I = 2/5mr^2 + mx^2 ?

20. May 26, 2015

haruspex

Good. Now what about the relationship between v, r, x and omega?