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--- book:chap3:3.6_problems [2021/11/28 12:57] – created abril
+++ book:chap3:3.6_problems [2022/01/30 04:23] (current) – jv
@@ Line 240: / Line 240: @@
 **c)** Indicate the direction and magnitude of the gradient by appropriate arrows in the sketch showing the contour lines. In which direction is the gradient pointing?
+----
+<wrap #quest_lineintegral_venyl-disc > Problem 3.22: </wrap>** Length of the groove of venyl records **
+\\
+<WRAP 200pt right >
+{{  :book:chap3:venyl-disc-dj_spooky.jpg?200  }}
+DJ Spooky playing venyl records at the Sundance Film Festival (2003),
+[[https://commons.wikimedia.org/wiki/File:Spooky.jpg|Eddie Codel (Ekai) via Wikimedia Commons]],
+[[http://creativecommons.org/licenses/by-sa/3.0/|CC BY-SA 3.0]]
+</WRAP>
+The sound information in venyl records is stored in small undulations of the walls of a groove
+that runs in a spiral from the outer rim of the disc, at a distance $R_o$ from the centre,
+towards its center where it stops at a inner radius $R_i < R_o$.
+Neighboring lanes of the groove have a fixed distance $d$.
+The sound is picked up by a needle that glides through the groove when the disc is turned.
+Let $\theta = 0$ be the angle where the needle first touched the disc at time $t_0$,
+and $\theta(t)$ be the overall traversed angle at time $t \geq t_0$.
+In the frame of the disc we will denote the position of the needle as
+\begin{align*} %  \label{eq:}
+    \mathbf q (\theta) = R(\theta) \: \hat{\mathbf r}(\theta)
+\end{align*}
+Here  $\hat{\mathbf r}(\theta)$ is the radial unit vector of polar coordinates taken with respect to the center of the disc,
+and $R(\theta)$ is the distance from the center.
+**a)**
+Verify that  $R(\theta) = R_a - \varepsilon \: \theta$.
+What is the relation between  $\varepsilon$ and $d$?
+**b)**
+What is the speed  $v$  of the needle while it glides through the groove?
+Show that one can express $v$ as follows
+\begin{align*} %  \label{eq:}
+   v = \dot\theta \; \sqrt{ R^2 + f(R) }
+\end{align*}
+Determine  $f(R)$.
+**★ c)**
+Demonstrate that the length $L$ of the groove that is traversed while turning from  $\theta_I$ to $\theta_E$
+can be written as
+\begin{align*} %  \label{eq:}
+   L  = \int_i^e \text{d}\theta \: \sqrt{1 + \theta^2}
+\end{align*}
+How do $i$ and $e$ depend on $\theta_I$ and $\theta_E$?
+** d)**
+Observe that
+\begin{align*} %  \label{eq:}
+      \sqrt{1 + \theta^2} = \frac{1}{\sqrt{1 + \theta^2}} +  \frac{x^2}{\sqrt{1 + \theta^2}}
+\end{align*}
+where the first term is the derivative of $\text{arcsinh}(x)$
+and the latter term is related to $\sqrt{1 + \theta^2}$ by partial integration.
+Use this information to evaluate $L$.
+**★ e)**
+Compare your result with the following estimate:
+   - The groove covers an area of size  $A = \pi \, R_o^2 - \pi \, R_i^2$.\\ Why does this hold? How are  $R_o$ and $R_i$  related to  $\theta_I$ und $\theta_E$?
+  - The area can also be estimated by multiplying the length $L$ of the groove with the groove distance, $d$, such that also $A \approx L\times d$.
+How well does the resulting estimate of $L$ agree with the result of the explicit calculation obtained in d).
+** f)**
+An LP has an outer radius of $R_o = 15\,\text{cm}$ and the groove stops at the inner radius of about $R_i = 8\,$cm.
+It is played with an angular speed of $33 \frac{1}{3}\,\text{rpm}$ (revolutions per minute),
+and each side is playing for about $22\,\text{min.}$
+What is the length of the groove and what is the distance $d$ between its neighboring revolutions?
 ==== 3.6.3 Transfer and Bonus Problems, Riddles ====