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Suppose we have a shell with mass $M$ and radius $R$. If we let that roll without slipping down a ramp of angle theta to the horizontal, we can easily find the acceleration of the shell the instant the ball is let go.

Now, what if we completely fill the shell of mass $M$ and radius $R$ with a frictionless fluid of mass $M$ and let it roll down (without slipping) a ramp with angle theta? I have been thinking of this for a while but cannot seem to find what the instantaneous acceleration is as soon as the $\text{shell + liquid}$ is let go.

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Is this a homework question? –  Dave Jan 15 '13 at 14:14
    
The mass moment of inertia will differ in the two cases. –  ja72 Aug 13 '13 at 18:04
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2 Answers 2

If your goal is to assume the fluid inside is friction-less, then consider a rotating hollow sphere with a non-rotating mass inside. Include the total mass of the shell and water in $m$, but only include the inertia from the shell in $I$. Secondly, if you want the acceleration then you can't rely on energy methods, and need to write a free body diagram in 2D. I've oriented the x-axis along the downward ramp direction, and the y-axis perpendicular to that.

  • $a_x$ acceleration of the object's center along the downward ramp [$m/s^2$],
  • $\alpha$ angular acceleration (about z-axis) of the object's center [$rad/s^2$],
  • $f_x$ friction force between shell and ramp (along x-axis pointed opposite of $a_x$) [$N$],
  • $m$ the total mass of the object (fluid+shell) [$kg$],
  • $g$ acceleration due to gravity [$m/s^2$],
  • $\theta$ angle between the ramp and the horizontal ground [$rad$],
  • $I$ moment of inertia about the center of mass for the shell [$kg \cdot m^2$],
  • $N=mg\text{cos}(\theta)$ normal force perpendicular to ramp surface (positive y-direction) [$N$]
  • $R$ outer radius of the shell,
  • $M_z$ moments about z-axis.

First solve for the friction force $f$ assuming that no slip occurs, implying the shell's angular acceleration is matched to the ball's center acceleration, $$a_x=\alpha R$$ Summing the moments (about z-axis) to solve for friction force $f$ $$\sum M_z = I\alpha = fR$$

$$f = \frac{Ia_x}{R^2}$$

For completeness here is the y-direction equation of motion, though it's not needed: $$\sum F_y= 0 = N - mg \cdot \text{cos}(\theta)$$ Next, create the x-direction equation of motion: $$\sum F_x = ma_x = mg \cdot \text{sin}(\theta)-f$$ Substitute in the previous $f$ and solve for $a_x$:

$$ a_x = \frac{mg \cdot \text{sin}\theta}{m+I/R^2}$$

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One way to approach this type of problem is to use conservation of energy

Let:

  • $m=2M$ be the total mass of the object (fluid+shell)
  • $g$ be the acceleration due to gravity,
  • $h$ be the vertical distance the object moved from start,
  • $v$ be the speed of the object down the ramp, and
  • $I$ be the moment of inertia about the center of mass for the shell.
  • $R$ be the radius of the shell.

I assume that the shell rolls without slipping, and that the fluid does not interact with the rolling shell. The latter is artificial, and probably could be alleviated, to first order, by increasing the effective moment of inertia of the shell and/or including a frictional force. Leaving these aside, we get

$mgh=\frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 $

$\omega = \frac{v}{R}$

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Thank you, but I'm particualrly concerened with the instantateous acceleration- that is, the acceleration of the shell+fluid as soon as wel let it go. How would one compute that? –  Brian Jan 15 '13 at 14:21
    
In other words, would adding a frictionless fluid change the inertia of the shell? Would it make it the inertia of a solid ball? –  Brian Jan 15 '13 at 18:01
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It would not make it the inertia of a solid ball, because the fluid would never be set in rotation if frictionless. On the other hand, in a solid ball, every mass element in the ball rotates in a collective movement around some common axis. Both these situations are different as far as the mechanics goes. –  Mathusalem Jun 14 '13 at 17:08
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