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Chapter 10: Problem 28

A sewing machine needle moves with a rapid vibratory motion, rather like SHM, as it sews a seam. Suppose the needle moves \(8.4 \mathrm{mm}\) from its highest to its lowest position and it makes 24 stitches in 9.0 s. What is the maximum needle speed?

Short Answer

Expert verified
Answer: The maximum speed of the sewing machine needle is 0.1408 m/s.

Step by step solution

01

Understanding the given information

We are given the following information: - Amplitude (A): The distance from the highest to the lowest position, which is \(8.4 \mathrm{mm}\) - Number of stitches: 24 - Time taken to perform the stitches (t): 9.0 s
02

Calculate the time period for one full oscillation

One full oscillation occurs when the needle completes one up-down cycle. The time period for one full oscillation (T) can be calculated by dividing the total time taken by the number of stitches as each stitch corresponds to a single oscillation: \(T=\frac{t}{\text{number of stitches}}\) Plug in the values: \(T = \frac{9.0 \mathrm{s}}{24}\) \(T = 0.375 \mathrm{s}\)
03

Determine the angular frequency

The angular frequency (\(\omega\)) can be calculated using the formula: \( \omega = \frac{2 \pi}{T} \) Plug in the value for T: \( \omega = \frac{2 \pi}{0.375 \mathrm{s}} \) \( \omega = 16.76 \ \text{s}^{-1} \)
04

Calculate the maximum needle speed

The maximum speed (vmax) of the needle in SHM can be found using the formula: \( \mathrm{v_{max}} = A \omega \) Plug in the values for A and \(\omega\): \( \mathrm{v_{max}} = (8.4 \mathrm{mm})(16.76 \ \text{s}^{-1})\) \( \mathrm{v_{max}} = 140.8 \ \text{mm} \ \text{s}^{-1} \)
05

Express the answer in the right unit

The maximum needle speed can be converted from \(\text{mm} \ \text{s}^{-1}\) to \(\text{m} \ \text{s}^{-1}\): \( \mathrm{v_{max}} = 140.8 \ \text{mm} \ \text{s}^{-1} \times \frac{1 \text{m}}{1000 \text{mm}} = 0.1408 \ \text{m} \ \text{s}^{-1}\) The maximum needle speed is \(0.1408 \ \text{m} \ \text{s}^{-1}\).

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Most popular questions from this chapter

A \(230.0-\mathrm{g}\) object on a spring oscillates left to right on a frictionless surface with a frequency of \(2.00 \mathrm{Hz}\). Its position as a function of time is given by \(x=(8.00 \mathrm{cm})\) sin \(\omega t\) (a) Sketch a graph of the elastic potential energy as a function of time. (b) The object's velocity is given by \(v_{x}=\omega(8.00 \mathrm{cm}) \cos \omega t .\) Graph the system's kinetic energy as a function of time. (c) Graph the sum of the kinetic energy and the potential energy as a function of time. (d) Describe qualitatively how your answers would change if the surface weren't frictionless.
(a) Sketch a graph of \(x(t)=A \sin \omega t\) (the position of an object in SHM that is at the equilibrium point at \(t=0\) ). (b) By analyzing the slope of the graph of \(x(t),\) sketch a graph of $v_{x}(t) .\( Is \)v_{x}(t)$ a sine or cosine function? (c) By analyzing the slope of the graph of \(v_{x}(t),\) sketch \(a_{x}(t)\) (d) Verify that \(v_{x}(t)\) is \(\frac{1}{4}\) cycle ahead of \(x(t)\) and that \(a_{x}(t)\) is \(\frac{1}{4}\) cycle ahead of \(v_{x}(t) .\) (W) tutorial: sinusoids)
A pendulum is made from a uniform rod of mass \(m_{1}\) and a small block of mass \(m_{2}\) attached at the lower end. (a) If the length of the pendulum is \(L\) and the oscillations are small, find the period of the oscillations in terms of \(m_{1}, m_{2}, L,\) and \(g .\) (b) Check your answer to part (a) in the two special cases \(m_{1} \gg m_{2}\) and \(m_{1}<<m_{2}\).
It takes a flea \(1.0 \times 10^{-3}\) s to reach a peak speed of $0.74 \mathrm{m} / \mathrm{s}$ (a) If the mass of the flea is \(0.45 \times 10^{-6} \mathrm{kg},\) what is the average power required? (b) Insect muscle has a maximum output of 60 W/kg. If \(20 \%\) of the flea's weight is muscle, can the muscle provide the power needed? (c) The flea has a resilin pad at the base of the hind leg that compresses when the flea bends its leg to jump. If we assume the pad is a cube with a side of \(6.0 \times 10^{-5} \mathrm{m},\) and the pad compresses fully, what is the energy stored in the compression of the pads of the two hind legs? The Young's modulus for resilin is $1.7 \times 10^{6} \mathrm{N} / \mathrm{m}^{2} .$ (d) Does this provide enough power for the jump?
Two steel plates are fastened together using four bolts. The bolts each have a shear modulus of \(8.0 \times 10^{10} \mathrm{Pa}\) and a shear strength of $6.0 \times 10^{8} \mathrm{Pa} .\( The radius of each bolt is \)1.0 \mathrm{cm} .$ Normally, the bolts clamp the two plates together and the frictional forces between the plates keep them from sliding. If the bolts are loose, then the frictional forces are small and the bolts themselves would be subject to a large shear stress. What is the maximum shearing force \(F\) on the plates that the four bolts can withstand? (IMAGE NOT COPY)
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