{"id":90871,"date":"2025-06-01T10:39:19","date_gmt":"2025-06-01T10:39:19","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90871"},"modified":"2025-06-01T10:39:19","modified_gmt":"2025-06-01T10:39:19","slug":"a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-%cf%89","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-%cf%89\/","title":{"rendered":"A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9"},"content":{"rendered":"

A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9t 10. The acceleration of the particle at time t = $\\frac{\\pi}{2\\omega}$ will be : (symbols here carry their usual meanings)<\/p>\n

[amp_mcq option1=”10 \u03c9” option2=”$-10\\omega^2$” option3=”0″ option4=”$\\frac{10}{\\omega}$” correct=”option3″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2023<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is 0.
\n<\/section>\n
\nThe given equation for the simple harmonic motion of a particle is $y = 10 \\cos \\omega t + 10$. This represents the displacement of the particle from a reference point (in this case, an origin shifted by 10 units). The velocity of the particle is the first derivative of displacement with respect to time: $v = \\frac{dy}{dt} = \\frac{d}{dt}(10 \\cos \\omega t + 10) = -10 \\omega \\sin \\omega t$. The acceleration of the particle is the first derivative of velocity with respect to time: $a = \\frac{dv}{dt} = \\frac{d}{dt}(-10 \\omega \\sin \\omega t) = -10 \\omega^2 \\cos \\omega t$. We need to find the acceleration at time $t = \\frac{\\pi}{2\\omega}$. Substituting this value of $t$ into the acceleration equation: $a\\left(t=\\frac{\\pi}{2\\omega}\\right) = -10 \\omega^2 \\cos\\left(\\omega \\cdot \\frac{\\pi}{2\\omega}\\right) = -10 \\omega^2 \\cos\\left(\\frac{\\pi}{2}\\right)$. Since $\\cos\\left(\\frac{\\pi}{2}\\right) = 0$, the acceleration at this time is $a = -10 \\omega^2 \\cdot 0 = 0$.
\n<\/section>\n
\nIn simple harmonic motion described by $y = A \\cos(\\omega t + \\phi) + C$, the term $A \\cos(\\omega t + \\phi)$ represents the oscillation about the equilibrium position. The acceleration is proportional to the displacement from the equilibrium position and directed towards it ($a = -\\omega^2 (y-C)$). In this case, the equilibrium position is at $y=10$. At $t = \\frac{\\pi}{2\\omega}$, the displacement $y = 10 \\cos(\\frac{\\pi}{2}) + 10 = 10(0) + 10 = 10$. Since the displacement is equal to the equilibrium position, the acceleration is zero, as expected.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9t 10. The acceleration of the particle at time t = $\\frac{\\pi}{2\\omega}$ will be : (symbols here carry their usual meanings) [amp_mcq option1=”10 \u03c9” option2=”$-10\\omega^2$” option3=”0″ option4=”$\\frac{10}{\\omega}$” correct=”option3″] This question was previously asked in UPSC CAPF – 2023 Download PDFAttempt Online … <\/p>\n

Detailed SolutionA simple harmonic motion of a particle is represented as, y = 10 cos \u03c9<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1105,1129,1128],"class_list":["post-90871","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1105","tag-mechanics","tag-physics","no-featured-image-padding"],"yoast_head":"\nA simple harmonic motion of a particle is represented as, y = 10 cos \u03c9<\/title>\n<meta name=\"description\" content=\"The correct answer is 0. The given equation for the simple harmonic motion of a particle is $y = 10 cos omega t + 10$. This represents the displacement of the particle from a reference point (in this case, an origin shifted by 10 units). The velocity of the particle is the first derivative of displacement with respect to time: $v = frac{dy}{dt} = frac{d}{dt}(10 cos omega t + 10) = -10 omega sin omega t$. The acceleration of the particle is the first derivative of velocity with respect to time: $a = frac{dv}{dt} = frac{d}{dt}(-10 omega sin omega t) = -10 omega^2 cos omega t$. We need to find the acceleration at time $t = frac{pi}{2omega}$. Substituting this value of $t$ into the acceleration equation: $aleft(t=frac{pi}{2omega}right) = -10 omega^2 cosleft(omega cdot frac{pi}{2omega}right) = -10 omega^2 cosleft(frac{pi}{2}right)$. Since $cosleft(frac{pi}{2}right) = 0$, the acceleration at this time is $a = -10 omega^2 cdot 0 = 0$.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-\u03c9\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9\" \/>\n<meta property=\"og:description\" content=\"The correct answer is 0. The given equation for the simple harmonic motion of a particle is $y = 10 cos omega t + 10$. This represents the displacement of the particle from a reference point (in this case, an origin shifted by 10 units). The velocity of the particle is the first derivative of displacement with respect to time: $v = frac{dy}{dt} = frac{d}{dt}(10 cos omega t + 10) = -10 omega sin omega t$. The acceleration of the particle is the first derivative of velocity with respect to time: $a = frac{dv}{dt} = frac{d}{dt}(-10 omega sin omega t) = -10 omega^2 cos omega t$. We need to find the acceleration at time $t = frac{pi}{2omega}$. Substituting this value of $t$ into the acceleration equation: $aleft(t=frac{pi}{2omega}right) = -10 omega^2 cosleft(omega cdot frac{pi}{2omega}right) = -10 omega^2 cosleft(frac{pi}{2}right)$. Since $cosleft(frac{pi}{2}right) = 0$, the acceleration at this time is $a = -10 omega^2 cdot 0 = 0$.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-\u03c9\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:39:19+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9","description":"The correct answer is 0. The given equation for the simple harmonic motion of a particle is $y = 10 cos omega t + 10$. This represents the displacement of the particle from a reference point (in this case, an origin shifted by 10 units). The velocity of the particle is the first derivative of displacement with respect to time: $v = frac{dy}{dt} = frac{d}{dt}(10 cos omega t + 10) = -10 omega sin omega t$. The acceleration of the particle is the first derivative of velocity with respect to time: $a = frac{dv}{dt} = frac{d}{dt}(-10 omega sin omega t) = -10 omega^2 cos omega t$. We need to find the acceleration at time $t = frac{pi}{2omega}$. Substituting this value of $t$ into the acceleration equation: $aleft(t=frac{pi}{2omega}right) = -10 omega^2 cosleft(omega cdot frac{pi}{2omega}right) = -10 omega^2 cosleft(frac{pi}{2}right)$. Since $cosleft(frac{pi}{2}right) = 0$, the acceleration at this time is $a = -10 omega^2 cdot 0 = 0$.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-\u03c9\/","og_locale":"en_US","og_type":"article","og_title":"A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9","og_description":"The correct answer is 0. The given equation for the simple harmonic motion of a particle is $y = 10 cos omega t + 10$. This represents the displacement of the particle from a reference point (in this case, an origin shifted by 10 units). The velocity of the particle is the first derivative of displacement with respect to time: $v = frac{dy}{dt} = frac{d}{dt}(10 cos omega t + 10) = -10 omega sin omega t$. The acceleration of the particle is the first derivative of velocity with respect to time: $a = frac{dv}{dt} = frac{d}{dt}(-10 omega sin omega t) = -10 omega^2 cos omega t$. We need to find the acceleration at time $t = frac{pi}{2omega}$. Substituting this value of $t$ into the acceleration equation: $aleft(t=frac{pi}{2omega}right) = -10 omega^2 cosleft(omega cdot frac{pi}{2omega}right) = -10 omega^2 cosleft(frac{pi}{2}right)$. Since $cosleft(frac{pi}{2}right) = 0$, the acceleration at this time is $a = -10 omega^2 cdot 0 = 0$.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-\u03c9\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:39:19+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-%cf%89\/","url":"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-%cf%89\/","name":"A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:39:19+00:00","dateModified":"2025-06-01T10:39:19+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is 0. The given equation for the simple harmonic motion of a particle is $y = 10 \\cos \\omega t + 10$. This represents the displacement of the particle from a reference point (in this case, an origin shifted by 10 units). The velocity of the particle is the first derivative of displacement with respect to time: $v = \\frac{dy}{dt} = \\frac{d}{dt}(10 \\cos \\omega t + 10) = -10 \\omega \\sin \\omega t$. The acceleration of the particle is the first derivative of velocity with respect to time: $a = \\frac{dv}{dt} = \\frac{d}{dt}(-10 \\omega \\sin \\omega t) = -10 \\omega^2 \\cos \\omega t$. We need to find the acceleration at time $t = \\frac{\\pi}{2\\omega}$. Substituting this value of $t$ into the acceleration equation: $a\\left(t=\\frac{\\pi}{2\\omega}\\right) = -10 \\omega^2 \\cos\\left(\\omega \\cdot \\frac{\\pi}{2\\omega}\\right) = -10 \\omega^2 \\cos\\left(\\frac{\\pi}{2}\\right)$. Since $\\cos\\left(\\frac{\\pi}{2}\\right) = 0$, the acceleration at this time is $a = -10 \\omega^2 \\cdot 0 = 0$.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-%cf%89\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-%cf%89\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/a-simple-harmonic-motion-of-a-particle-is-represented-as-y-10-cos-%cf%89\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"A simple harmonic motion of a particle is represented as, y = 10 cos \u03c9"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90871","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=90871"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90871\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=90871"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=90871"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=90871"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}