# Useful Derivation for Labs Involving Rolling Balls Down an Inclined Plane

• MHB
• Ackbach
In summary, the conversation discusses the problem of large experimental errors in beginning mechanics physics labs and how it can be caused by factors such as friction and comparing apples to oranges. The theory of balls rolling down inclines is also discussed, with the solution being derived using conservation of energy and rotational kinetic energy. A simplified version of the solution is also presented, which involves differentiation instead of integration. The idea of using a general moment of inertia and finding that only the shape of the object is significant in rolling under gravity is also mentioned.
My preferred solution would be:

We have an accelerating force of ##F = mg\sin \theta## down the slope. The work done by this force over a distance ##x## is ##Fx## and this must equal the total linear and rotational KE. Hence$$mgx \sin \theta = \frac 1 2mv^2 + \frac 1 2 Iw^2 = \frac 1 2 mv^2 + \frac 1 2 kmr^2\frac {v^2}{r^2} = \frac 1 2 mv^2(1 +k)$$This gives us $$v^2 = 2\frac{g\sin \theta}{1 +k}x$$And using $$v^2 = 2ax$$gives$$a = \frac{g\sin \theta}{1 +k}$$Finally, taking ##k = \frac 2 5## and using $$x = \frac 1 2 at^2$$ we get$$x = \frac{5g\sin \theta }{14}\ t^2$$And that only uses what I believe to be high school physics and maths.

This has the disadvantage that it assumes ##v_0=0##, i.e., it's not the general solution. Instead of forcing people memorize SUVAT equations one should teach them how to derive them from the general Newtonian laws of motion, and this indeed can be done with high-school mathematics only as demonstrated in my previous posting above.

vanhees71 said:
This has the disadvantage that it assumes ##v_0=0##, i.e., it's not the general solution.
A change of reference frame takes care of that.
vanhees71 said:
Instead of forcing people memorize SUVAT equations one should teach them how to derive them from the general Newtonian laws of motion, and this indeed can be done with high-school mathematics only as demonstrated in my previous posting above.
A physics student needs to be able remember some things. Do you advocate deriving the equations of electromagnetism from first principles every time you need them?

malawi_glenn
Different people have different ideas on what is clear, what is straight-forward, etc. Anyone should feel free to use their own derivation in place of mine.

PS if the student can't "memorise" that ##KE = \frac 1 2 mv^2## etc. then physics will be a long haul!

malawi_glenn
PeroK said:
A change of reference frame takes care of that.

A physics student needs to be able remember some things. Do you advocate deriving the equations of electromagnetism from first principles every time you need them?
Of course not, but just rote learning some equations without understanding them is also contrary to what we really want, right?

Ackbach said:
The derivation is useful in the lab. That is, it is useful for the derivation to have been accomplished so that, in the lab, you get far less experimental error. The method of deriving the result is basically not the point of the OP at all. I derive it so that the student has confidence in the result, but that's all. The point of the OP is that having the correct formula reduces your experimental error.
The result is useful of course.

vanhees71 said:
This has the disadvantage that it assumes ##v_0=0##, i.e., it's not the general solution. Instead of forcing people memorize SUVAT equations one should teach them how to derive them from the general Newtonian laws of motion, and this indeed can be done with high-school mathematics only as demonstrated in my previous posting above.
Will be a more cumbersome lab to conduct, you need an additional setup of speed measurement if you do not release the cylinder/ball/thing at rest. I hate photogates!

vanhees71
vanhees71 said:
Of course not, but just rote learning some equations without understanding them is also contrary to what we really want, right?
Yes, but just because the student remembers something doesn't mean they don't understand what they are doing.

vanhees71 and malawi_glenn
For example, I remember that ##v^2-u^2 = 2as,## is really useful in some cases that may otherwise be problematic. I remember that that is a formula not to forget! Memory is fundamental to thinking IMHO.

malawi_glenn and vanhees71
PeroK said:
For example, I remember that ##v^2-u^2 = 2as,## is really useful in some cases that may otherwise be problematic. I remember that that is a formula not to forget! Memory is fundamental to thinking IMHO.
I refer to this forumlas as the "ass" formula, helps the students remember it

vanhees71
malawi_glenn said:
I refer to this forumlas as the "ass" formula, helps the students remember it
In case I forget it a derivation is:
$$v^2-u^2 = (v - u)(v + u)= at(v +u)= 2at\frac{v +u} 2 = 2a(tv_{avg}) = 2as$$It's simpler, of course, once you know about energy!

Last edited:
I suspect @malawi_glenn prefers the derivation
$$a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx} \quad \Rightarrow \quad \int_{v_0}^v v\,dv = \int_{x_0}^x a\,dx \quad \Rightarrow \quad v^2 - v_0^2 = 2a(x-x_0)$$

vanhees71 and malawi_glenn
Nah as long as you know that a is constant, peroK derivation is preferrable since it does not require calculus. My students however prefers your derivation because they think using (u+v)/2 = vtavg is magic

Earlier, I did teach that if we have ##v^2 = kx## then we can just use ##v^2 - v_0{}^2 = 2a \Delta x## formula to figure out ##a##. But students did not perform well, and there was lot of debate why we could do that, since ##v^2 - v_0{}^2 = 2a \Delta x## is only valid for constant acceleration and it was not evident for them why ##v^2 = kx## is such case.

Same with the ##s = v_0t + \dfrac{at^2}{2}## formula, before they learned how to differentiate and integrate, this formula was magic to them. Counting areas in a vt-diagram they thought was very tricky.

Last edited:
vanhees71 and PeroK
To avoid calculus is misleading since calculus makes the subject easier. That's why nowadays we don't use the methods of Newton's principia anymore but Euler's formulation in terms of calculus.

Ackbach and malawi_glenn
vanhees71 said:
To avoid calculus is misleading since calculus makes the subject easier. That's why nowadays we don't use the methods of Newton's principia anymore but Euler's formulation in terms of calculus.
Yeah but in ours, and most schools, syllabus, physics courses starts prior to calculus based math classes...

pbuk and PeroK
In any case, basic algebra, geometry and trigonometry must come before calculus.

vanhees71 and malawi_glenn