Question

You have a summer internship, working with other interns on an archeological dig. Your intern team has found a perfectly cylindrical object of an unknown material. Examination of the visible surface shows that the composition of the object seems to be uniform. The object has a mass of 15.7 kg and a radius of 5.00 cm. The lead archeologist wants to know if the artifact is hollow, but the x-ray machine and other scanning equipment have broken down, so there is no way to look inside. Your team comes up with the idea of building U-shaped supports from wood and laying the cylinder horizontally between the supports. The wood can be sanded and oiled to almost eliminate friction. In this way, the cylindrical artifact is free to rotate around its long, horizontal axis. You wrap a long piece of twine several times around the cylinder and attach a 2.00-kg pickax to the free hanging end of the twine. When the pickax is released from rest, it descends and causes the cylinder to rotate. (a) You measure the falling of the pickax and find that it falls 1.50 m in 1.45 s. Is the cylinder hollow? (b) Suppose you measure the falling of the pickax through the same distance and find it to take 1.13 s. What can you conclude about the cylinder now?

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From equation 10.11, the relation between the angular acceleration $\color{#c34632}\alpha$ of a rotating rigid object to the tangential acceleration $\color{#c34632}a_t$ of a point at distance $\color{#c34632}r$ from the center of the rotation motion is given by:

at=rα\begin{gather*} a_t = r\alpha\tag{1} \end{gather*}

From equation 10.14, the magnitude of the torque $\color{#c34632}\tau$ associated with a force $\color{#c34632}\vec{\textbf{F}}$	acting on an object at a distance $\color{#c34632}r$ from the rotation axis is given by:

τ=rFsinϕ\begin{gather*} \tau = rF\sin{\phi}\tag{2} \end{gather*}

From equation 10.18, if a rigid object of moment of inertia $\color{#c34632}I$ is free to rotate about a fixed axis has a net external torque $\color{#c34632}\tau_{\text{ext}}$ acting on it, the object undergoes an angular acceleration $\color{#c34632}\alpha$, where:

τext=Iα\begin{gather*} \tau_{\text{ext}} = I \alpha\tag{3} \end{gather*}

From table 10.2, the moment of inertia of a solid cylinder of mass $\color{#c34632}M$ and radius $\color{#c34632}R$ rotating about its axis is given by:

I=12MR2\begin{gather*} I = \dfrac{1}{2}MR^2\tag{4} \end{gather*}

And the moment of inertia of a hollow cylinder of mass $\color{#c34632}M$, inner radius $\color{#c34632}R_1$ and outer radius $\color{#c34632}R_2$ and rotating about its axis is given by:

I=12M(R12+R22)\begin{gather*} I = \dfrac{1}{2}M(R_1^2 + R_2^2)\tag{5} \end{gather*}

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