{"id":2024,"date":"2025-07-31T23:54:16","date_gmt":"2025-07-31T23:54:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2024"},"modified":"2025-10-16T19:08:36","modified_gmt":"2025-10-16T19:08:36","slug":"modeling-with-trigonometric-equations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/modeling-with-trigonometric-equations-learn-it-2\/","title":{"raw":"Modeling with Trigonometric Equations: Learn It 2","rendered":"Modeling with Trigonometric Equations: Learn It 2"},"content":{"raw":"

Modeling Harmonic Motion<\/h2>\r\nHarmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion<\/strong> applications cycle through their periods with no outside interference, harmonic motion<\/strong> requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.\r\n

Simple Harmonic Motion<\/h3>\r\nA type of motion described as simple harmonic motion<\/strong> involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When [latex]t=0,d=0[\/latex].\r\n\r\n
\r\n

simple harmonic<\/h3>\r\nWe see that simple harmonic motion equations are given in terms of displacement:\r\n\r\n[latex]d=a\\cos \\left(\\omega t\\right)\\text{ or }d=a\\sin \\left(\\omega t\\right)[\/latex]\r\n\r\nwhere [latex]|a|[\/latex] is the amplitude, [latex]\\frac{2\\pi }{\\omega }[\/latex] is the period, and [latex]\\frac{\\omega }{2\\pi }[\/latex] is the frequency, or the number of cycles per unit of time.\r\n\r\n<\/section>
For the given functions,\r\n
    \r\n \t
  1. Find the maximum displacement of an object.<\/li>\r\n \t
  2. Find the period or the time required for one vibration.<\/li>\r\n \t
  3. Find the frequency.<\/li>\r\n \t
  4. Sketch the graph.\r\n
      \r\n \t
    1. [latex]y=5\\sin \\left(3t\\right)[\/latex]<\/li>\r\n \t
    2. [latex]y=6\\cos \\left(\\pi t\\right)[\/latex]<\/li>\r\n \t
    3. [latex]y=5\\cos \\left(\\frac{\\pi }{2}t\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"65116\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"65116\"]\r\n
        \r\n \t
      1. [latex]y=5\\sin \\left(3t\\right)[\/latex]\r\n
          \r\n \t
        1. The maximum displacement is equal to the amplitude, [latex]|a|[\/latex], which is 5.<\/li>\r\n \t
        2. The period is [latex]\\frac{2\\pi }{\\omega }=\\frac{2\\pi }{3}[\/latex].<\/li>\r\n \t
        3. The frequency is given as [latex]\\frac{\\omega }{2\\pi }=\\frac{3}{2\\pi }[\/latex].<\/li>\r\n \t
        4. The graph indicates the five key points.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
        5. [latex]y=6\\cos \\left(\\pi t\\right)[\/latex]\r\n
            \r\n \t
          1. The maximum displacement is [latex]6[\/latex].<\/li>\r\n \t
          2. The period is [latex]\\frac{2\\pi }{\\omega }=\\frac{2\\pi }{\\pi }=2[\/latex].<\/li>\r\n \t
          3. The frequency is [latex]\\frac{\\omega }{2\\pi }=\\frac{\\pi }{2\\pi }=\\frac{1}{2}[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
          4. [latex]y=5\\cos \\left(\\frac{\\pi }{2}\\right)t[\/latex]\r\n
              \r\n \t
            1. The maximum displacement is [latex]5[\/latex].<\/li>\r\n \t
            2. The period is [latex]\\frac{2\\pi }{\\omega }=\\frac{2\\pi }{\\frac{\\pi }{2}}=4[\/latex].<\/li>\r\n \t
            3. The frequency is [latex]\\frac{1}{4}[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
              \r\n

              A weight on a spring oscillates vertically with position given by [latex]y = 7\\cos\\left(\\frac{\\pi}{3}t\\right)[\/latex] cm, where [latex]t[\/latex] is time in seconds. Find:<\/p>\r\n\r\n

                \r\n \t
              1. The maximum displacement of the weight<\/li>\r\n \t
              2. The period of oscillation<\/li>\r\n \t
              3. The frequency<\/li>\r\n \t
              4. At what time during the first period does the weight first reach [latex]y = 3.5[\/latex] cm?<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"

                Modeling Harmonic Motion<\/h2>\n

                Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion<\/strong> applications cycle through their periods with no outside interference, harmonic motion<\/strong> requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.<\/p>\n

                Simple Harmonic Motion<\/h3>\n

                A type of motion described as simple harmonic motion<\/strong> involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When [latex]t=0,d=0[\/latex].<\/p>\n

                \n

                simple harmonic<\/h3>\n

                We see that simple harmonic motion equations are given in terms of displacement:<\/p>\n

                [latex]d=a\\cos \\left(\\omega t\\right)\\text{ or }d=a\\sin \\left(\\omega t\\right)[\/latex]<\/p>\n

                where [latex]|a|[\/latex] is the amplitude, [latex]\\frac{2\\pi }{\\omega }[\/latex] is the period, and [latex]\\frac{\\omega }{2\\pi }[\/latex] is the frequency, or the number of cycles per unit of time.<\/p>\n<\/section>\n

                For the given functions,<\/p>\n
                  \n
                1. Find the maximum displacement of an object.<\/li>\n
                2. Find the period or the time required for one vibration.<\/li>\n
                3. Find the frequency.<\/li>\n
                4. Sketch the graph.\n
                    \n
                  1. [latex]y=5\\sin \\left(3t\\right)[\/latex]<\/li>\n
                  2. [latex]y=6\\cos \\left(\\pi t\\right)[\/latex]<\/li>\n
                  3. [latex]y=5\\cos \\left(\\frac{\\pi }{2}t\\right)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n