Another nice problem from Savchenko

[wrobel]

A slab rests on two massless rollers of different radii such that the angle of the slab's inclination is $\alpha$. There is no slipping between the slab and the rollers, nor between the rollers and the ground. Prove that the slab moves with purely translational motion and find its acceleration.
(the value of the acceleration is $g\sin(\alpha/2)$)
The standard obstacle here is finding the reactions of the rollers, but the use of the energy integral allows one to avoid this difficulty. Accurate proof that the motion is translational also requires some kinematic techniques.

It is also interesting to note that the hypothesis of ideal constraints here does not imply that the reactions of the rollers are perpendicular to the slab. This is not the case, even though the rollers are massless. It is because the net reaction is perpendicular to the virtual displacement, as it must be

 

[anuttarasammyak]

I would try some math.

1. I observe that angle $\alpha$ does not change during the motion but the slab moves with NOT purely (horizontal, I would add. See my post #6. ) translational motion.　Displacement d of the slab, which has horizontal and perpendicular components, is
$$d=r\theta\sqrt{(1+\cos\alpha)^2+\sin^2\alpha}=2 r\theta\cos\frac{\alpha}{2}$$
where r, $\theta$ are radius and rotation angle of either one of the rollers. Thus
$$\vec{d}=d(\cos\frac{\alpha}{2},-\sin\frac{\alpha}{2})$$
and
$$\vec{g} \cdot \frac{\vec{d}}{d} = g \sin\frac{\alpha}{2}$$
Therefore
$$d = \frac{1}{2}g \sin\frac{\alpha}{2} \ t^2+\dot{d}_0t+d_0$$

2. Lagrangian for massless rollers case is
$$L=m[\frac{1}{2} (2 r\dot{\theta}\cos\frac{\alpha}{2})^2 + g r\theta \sin\alpha)]=4mr^2\cos^2\frac{\alpha}{2}(\ \frac{1}{2}\dot\theta^2+\frac{g}{2r}\tan\frac{\alpha}{2}\ \theta\ )$$
where m is mass of the slab. By solving the Lagrange equation
$$ \ddot{\theta}=\frac{g}{2r}\tan\frac{\alpha}{2}$$
Thus
$$\ddot{d}=g \sin\frac{\alpha}{2}$$

 

[wrobel]

My version is
$$L=mr^2\dot\theta^2(1+\cos\alpha)+mgr\theta\sin\alpha$$
hopefully that is the same:)

 

[wrobel]

angle α does not change during the motion but the slab moves with NOT purely translational motion

?????
By definition, purely translational motion is when all the points of a rigid body move with the same velocity

 

[anuttarasammyak]

The drop of the slab releases potential energy which enhances kinetic energy. The perpendicular component of the slab displacement should be considered as well as the horizontal component.

 

[anuttarasammyak]

By definition, purely translational motion is when all the points of a rigid body move with the same velocity

My bad. I confused translational with horizontal translational.