book:chap3:3.6_problems
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**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? | **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? | ||
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+ | <wrap # | ||
+ | \\ | ||
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+ | <WRAP 200pt right > | ||
+ | {{ : | ||
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+ | DJ Spooky playing venyl records at the Sundance Film Festival (2003), | ||
+ | [[https:// | ||
+ | [[http:// | ||
+ | </ | ||
+ | |||
+ | 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 | ||
+ | |||
+ | **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 | ||
+ | |||
+ | **★ 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:} | ||
+ | | ||
+ | \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\, | ||
+ | It is played with an angular speed of $33 \frac{1}{3}\, | ||
+ | and each side is playing for about $22\, | ||
+ | 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 ==== | ==== 3.6.3 Transfer and Bonus Problems, Riddles ==== |
book/chap3/3.6_problems.1638100628.txt.gz · Last modified: 2021/11/28 12:57 by abril