{"id":147631,"date":"2024-10-03T06:29:57","date_gmt":"2024-10-03T06:29:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=147631"},"modified":"2024-10-03T06:30:00","modified_gmt":"2024-10-03T06:30:00","slug":"the-overall-length-of-a-piccolo-is-32-0-cm","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/03\/the-overall-length-of-a-piccolo-is-32-0-cm\/","title":{"rendered":"The overall length of a piccolo is 32.0 cm"},"content":{"rendered":"\n<p>The overall length of a piccolo is 32.0 cm. The resonating air column vibrates as in a pipe that is open at both ends. (a) Find the frequency of the lowest note a piccolo can play. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is 4,000 Hz, find the distance between adjacent antinodes for this mode of vibration.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To analyze the piccolo&#8217;s resonance and sound properties, we can use the following concepts from wave physics related to open pipes.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(a) Frequency of the Lowest Note<\/h3>\n\n\n\n<p>For a pipe open at both ends, the fundamental frequency (first harmonic) can be calculated using the formula:<\/p>\n\n\n\n<p>[<br>f = \\frac{v}{\\lambda}<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( f ) = frequency of the sound<\/li>\n\n\n\n<li>( v ) = speed of sound in air (approximately 343 m\/s at room temperature)<\/li>\n\n\n\n<li>( \\lambda ) = wavelength<\/li>\n<\/ul>\n\n\n\n<p>In a pipe that is open at both ends, the wavelength of the fundamental frequency is twice the length of the pipe. Thus:<\/p>\n\n\n\n<p>[<br>\\lambda = 2L<br>]<\/p>\n\n\n\n<p>For a piccolo with a length ( L = 32.0 \\, \\text{cm} = 0.32 \\, \\text{m} ):<\/p>\n\n\n\n<p>[<br>\\lambda = 2 \\times 0.32 \\, \\text{m} = 0.64 \\, \\text{m}<br>]<\/p>\n\n\n\n<p>Now, substituting ( \\lambda ) back into the frequency formula:<\/p>\n\n\n\n<p>[<br>f = \\frac{343 \\, \\text{m\/s}}{0.64 \\, \\text{m}} \\approx 535.16 \\, \\text{Hz}<br>]<\/p>\n\n\n\n<p>Therefore, the frequency of the lowest note a piccolo can play is approximately <strong>535 Hz<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(b) Distance Between Adjacent Antinodes for 4,000 Hz<\/h3>\n\n\n\n<p>For the highest note (4,000 Hz), we again use the wave relationship. The wavelength for this frequency can be calculated as follows:<\/p>\n\n\n\n<p>[<br>\\lambda = \\frac{v}{f} = \\frac{343 \\, \\text{m\/s}}{4000 \\, \\text{Hz}} \\approx 0.08575 \\, \\text{m}<br>]<\/p>\n\n\n\n<p>In a standing wave, there are specific points called nodes (points of no displacement) and antinodes (points of maximum displacement). The distance between adjacent antinodes is half of the wavelength:<\/p>\n\n\n\n<p>[<br>\\text{Distance between adjacent antinodes} = \\frac{\\lambda}{2} = \\frac{0.08575 \\, \\text{m}}{2} \\approx 0.042875 \\, \\text{m} \\approx 4.29 \\, \\text{cm}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>In summary, the lowest frequency of a piccolo is approximately <strong>535 Hz<\/strong>, and the distance between adjacent antinodes at the highest note of <strong>4,000 Hz<\/strong> is approximately <strong>4.29 cm<\/strong>. These calculations illustrate the relationship between pipe length, frequency, and wavelength, which are essential for understanding how musical instruments produce sound.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The overall length of a piccolo is 32.0 cm. The resonating air column vibrates as in a pipe that is open at both ends. (a) Find the frequency of the lowest note a piccolo can play. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-147631","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/147631","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=147631"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/147631\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=147631"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=147631"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=147631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}