Where does it getting wrong , when using $v^2 - u^2 = 2as $ down the incline, for different object having different moment of inertia?

Question

Well, Consider a situation there is a sphere and a ring, of same mass $M$ and radius $R$. They both starts rolling down the inclined plane. We know moments of them as well, $$I_\text{sphere}=\frac{2}{5}MR^2$$ and $$I_\text{ring}=MR^2$$ respectively. So, We know that sphere will have more transitional kinetic energy, so more velocity, so it will take less time to reach at bottom.

The question is while using equation for both, $$v^2 - u^2 = 2as, $$ initial velocity is $0$ for both, final velocity are different for both, but acceleration and distance traveled same. So, where is the blunder happening?

And also the equation $$v=u+at,$$ if velocity for sphere is greater, then what about the time? Why is the time taken less? Where are the equations getting wrong or is it me getting it wrong?

Answer 1

What you have left in calculations is acceleration,, a≠g nor a=gsinα. $$a= gsinα-F/M$$ where α is angle of incline, F is force of friction and M remains mass.. Since friction acting on both are different, their acceleration are different for same distance s.

Same goes for your second equation v=u+at, here a is different. Same goes for t, dont you think one with faster translation kinetic energy reach sooner? Hope this helps..

Answer 2

Your kinematic equation of $v^2-u^2=2as$ is correct, but just like your question here you are neglecting the effects of friction, which gives rise to different accelerations for each object.

Considering the net force acting on each object, we actually have two forces with components acting down the ramp: gravity ($mg\sin\theta$) and friction ($f$). Without friction, the objects only will have the force $mg\sin\theta$ acting down the ramp, and there would be no net torque acting about the center of each object. Therefore, each object would slide without rolling down the ramp with the same acceleration and reach the bottom of the incline at the same time!

So, what you need to do is determine the net force acting on each object: $$F_{net}=mg\sin\theta-f$$

However, just like the question of yours I referred to, by imposing the rolling without slipping condition, $a=\alpha R$, you are constraining friction to be a certain value for each object that depends on their moment of inertia $I$. This can be seen by considering the net torque on each object: $$\tau_{net}=I\alpha=\frac aRI=fR$$

So we see that in order to have rolling without slipping it must be the case that $$f=\frac{aI}{R^2}=\gamma ma$$ for $I=\gamma mR^2$

So we see that we end up with different frictional force for each object. Putting it all together: $$mg\sin\theta-\gamma ma=ma$$ $$a=\frac{g\sin\theta}{1+\gamma}$$

Showing what you already knew: the larger value of $\gamma$ causes a lower acceleration, and hence a longer time down the ramp when both objects roll without slipping down the ramp.

Answer 3

I get this solution:

The equations of motion are:

$I\,\ddot{\vartheta}=F_c\,R\qquad (1)$

$M\,\ddot{s}=-F_c\,R +M\,g\,\sin(\alpha)\qquad (2)$

and rolling without slipping

$\ddot{s}=\ddot{\vartheta}\,R\qquad (3)$

with $F_c$ is constraint force.

We have 3 equations for 3 unknowns $\ddot{s}\,,\ddot{\vartheta}$ and the constraint force $F_c$.

we get for the incline acceleration:

$\ddot{s}=\frac{M\,R^2\,g\,\sin(\alpha)}{M\,R^2+I}$

with $I_s=\frac{2}{5}\,M\,R^2$ and $I_r=M\,R^2$ we get:

$\frac{\ddot{s}_{\text{sphere}}}{\ddot{s}_{\text{ring}}}=\frac{10}{7}$

so the incline acceleration of the sphere is $\frac{10}{7}$ greater then the incline acceleration of the ring.