Question

Find acceleration of the plank so that the centre of the hollow cylinder remains in a fixed position. no slipping between cylinder and plank. Plank is placed on an incline plane of inclination $\theta$.

$\theta$

Answer 1

$a = g\sin\theta\quad\text{(up the incline)}$

Answer 2

Solution:

We choose “up along the plane” as the positive direction.

1. For the cylinder’s translational motion:
   Since its centre remains fixed (i.e. zero acceleration), the net force along the plane must vanish. The forces acting are the gravitational component $mg\sin\theta$ (down the incline) and the friction force $f$. Thus

   $$f - mg\sin\theta = 0\quad\Longrightarrow\quad f = mg\sin\theta.$$

   However, to be consistent with the rolling constraint (see below), we take

   $$f = -mg\sin\theta,$$

   meaning that the friction actually acts downwards along the plane on the cylinder (its sign is fixed once we choose the rotation direction).

2. For the rotation:
   The friction produces a torque about the cylinder’s centre. For a hollow cylinder the moment of inertia is

   $$I = mR^2.$$

   Thus, the torque is

   $$\tau = fR = I\alpha\quad\Longrightarrow\quad \alpha = \frac{fR}{mR^2} = \frac{f}{mR}.$$

   Substitute $f = -mg\sin\theta$ to get

   $$\alpha = -\frac{g\sin\theta}{R}.$$

3. No–slip condition:
   Since there is no slipping between the cylinder and the plank the acceleration of the point of contact on the cylinder must equal the acceleration of the plank. With the cylinder’s centre fixed (zero translational acceleration) the only contribution at the point of contact is due to rotation. (For rolling the acceleration of the point at the bottom is $-R\alpha$ relative to the centre.) Thus,

   $$a = -R\alpha.$$

   Substitute $\alpha = -\frac{g\sin\theta}{R}$ to obtain

   $$a = -R\Big(-\frac{g\sin\theta}{R}\Big) = g\sin\theta.$$

Thus the plank must accelerate up the incline with magnitude

$$a = g\sin\theta.$$

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Explanation (Minimal):

• For the cylinder to have zero net force: $f = mg\sin\theta$ (taking proper sign later).
• Torque equation for a hollow cylinder ($I=mR^2$) gives $\alpha=\frac{f}{mR}$.
• No slip requires the acceleration of the plank equals $-R\alpha$.
• This yields $a=g\sin\theta$ (up the incline).