{"id":1796,"date":"2025-07-28T19:39:53","date_gmt":"2025-07-28T19:39:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1796"},"modified":"2025-08-13T03:03:29","modified_gmt":"2025-08-13T03:03:29","slug":"arcs-and-sectors-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arcs-and-sectors-learn-it-3\/","title":{"raw":"Arcs and Sectors: Learn It 3","rendered":"Arcs and Sectors: Learn It 3"},"content":{"raw":"
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Use Linear and Angular Speed to Describe Motion on a Circular Path<\/h2>\r\n<\/section><\/section>In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed<\/strong> is speed along a straight path and can be determined by the distance it moves along (its displacement<\/strong>) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or [latex]10\\pi [\/latex] inches, every second. So the linear speed of the point is [latex]10\\pi [\/latex] in.\/s. The equation for linear speed is as follows where [latex]v[\/latex] is linear speed, [latex]s[\/latex] is displacement, and [latex]t[\/latex]\r\nis time.\r\n
[latex]v=\\frac{s}{t}[\/latex]<\/div>\r\nAngular speed<\/strong> results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as [latex]\\frac{360\\text{ degrees}}{4\\text{ seconds}}=[\/latex] 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where [latex]\\omega [\/latex] (read as omega) is angular speed, [latex]\\theta [\/latex] is the angle traversed, and [latex]t[\/latex] is time.\r\n
[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/div>\r\n
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<\/div>\r\n
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angular speed<\/h3>\r\nAs a point moves along a circle of radius [latex]r[\/latex], its angular speed<\/strong>, [latex]\\omega [\/latex], is the angular rotation [latex]\\theta [\/latex] per unit time, [latex]t[\/latex].\r\n

[latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/p>\r\n\r\n<\/section><\/div>\r\n<\/div>\r\nCombining the definition of angular speed with the arc length equation, [latex]s=r\\theta [\/latex], we can find a relationship between angular and linear speeds. The angular speed equation can be solved for [latex]\\theta [\/latex], giving [latex]\\theta =\\omega t[\/latex]. Substituting this into the arc length equation gives:\r\n

[latex]\\begin{align}s&=r\\theta \\\\ &=r\\omega t \\end{align}[\/latex]<\/div>\r\nSubstituting this into the linear speed equation gives:\r\n
[latex]\\begin{align} v&=\\frac{s}{t} \\\\ &=\\frac{r\\omega t}{t} \\\\ &=r\\omega \\end{align}[\/latex]<\/div>\r\n
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linear speed<\/h3>\r\nThe linear speed<\/strong>. [latex]v[\/latex], of the point can be found as the distance traveled, arc length [latex]s[\/latex], per unit time, [latex]t[\/latex].\r\n

[latex]v=\\frac{s}{t}[\/latex]<\/p>\r\nWhen the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation\r\n

[latex]v=r\\omega[\/latex]<\/p>\r\nThis equation states that the angular speed in radians, [latex]\\omega [\/latex], representing the amount of rotation occurring in a unit of time, can be multiplied by the radius [latex]r[\/latex] to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.\r\n\r\n<\/section>

How To: Given the amount of angle rotation and the time elapsed, calculate the angular speed.\r\n<\/strong>\r\n
    \r\n \t
  1. If necessary, convert the angle measure to radians.<\/li>\r\n \t
  2. Divide the angle in radians by the number of time units elapsed: [latex]\\omega =\\frac{\\theta }{t}[\/latex].<\/li>\r\n \t
  3. The resulting speed will be in radians per time unit.<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n
    A water wheel completes 1 rotation every 5 seconds. Find the angular speed in radians per second.\"Illustration[reveal-answer q=\"602649\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"602649\"]\r\n\r\nThe wheel completes 1 rotation, or passes through an angle of [latex]2\\pi [\/latex]\u00a0radians in 5 seconds, so the angular speed would be [latex]\\omega =\\frac{2\\pi }{5}\\approx 1.257[\/latex] radians per second.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.[reveal-answer q=\"737350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"737350\"][latex]\\frac{-3\\pi }{2}[\/latex]\u00a0rad\/s[\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]155235[\/ohm_question]<\/section>
    How To: Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed.\r\n<\/strong>\r\n
      \r\n \t
    1. Convert the total rotation to radians if necessary.<\/li>\r\n \t
    2. Divide the total rotation in radians by the elapsed time to find the angular speed: apply [latex]\\omega =\\frac{\\theta }{t}[\/latex].<\/li>\r\n \t
    3. Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply [latex]v=r\\mathrm{\\omega}[\/latex].Example 11: Finding a Linear Speed<\/li>\r\n<\/ol>\r\n<\/section><\/section>
      A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.[reveal-answer q=\"461618\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"461618\"]Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road.\r\n\r\nWe begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:\r\n

      [latex]180\\frac{\\cancel{\\text{rotations}}}{\\text{minute}}\\cdot \\frac{2\\pi \\text{radians}}{\\cancel{\\text{rotation}}}=360\\pi \\frac{\\text{radians}}{\\text{minute}}[\/latex]<\/p>\r\nUsing the formula from above along with the radius of the wheels, we can find the linear speed:\r\n

      [latex]\\begin{align} v&=\\left(14\\text{inches}\\right)\\left(360\\pi \\frac{\\text{radians}}{\\text{minute}}\\right) \\\\ &=5040\\pi \\frac{\\text{inches}}{\\text{minute}} \\end{align}[\/latex]<\/p>\r\nRemember that radians are a unitless measure, so it is not necessary to include them.\r\n\r\nFinally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.\r\n

      [latex]5040\\pi \\frac{\\cancel{\\text{inches}}}{\\cancel{\\text{minute}}}\\cdot \\frac{\\text{1}\\cancel{\\text{feet}}}{\\text{12}\\cancel{\\text{inches}}}\\cdot \\frac{\\text{1 mile}}{\\text{5280}\\cancel{\\text{feet}}}\\cdot \\frac{\\text{60}\\cancel{\\text{minutes}}}{\\text{1 hour}}\\approx 14.99\\text{miles per hour (mph)}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      A satellite is rotating around Earth at 0.25 radians per hour at an altitude of 242 km above Earth. If the radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.[reveal-answer q=\"851398\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"851398\"]1655 kilometers per hour[\/hidden-answer]<\/section>","rendered":"
      \n
      \n

      Use Linear and Angular Speed to Describe Motion on a Circular Path<\/h2>\n<\/section>\n<\/section>\n

      In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed<\/strong> is speed along a straight path and can be determined by the distance it moves along (its displacement<\/strong>) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or [latex]10\\pi[\/latex] inches, every second. So the linear speed of the point is [latex]10\\pi[\/latex] in.\/s. The equation for linear speed is as follows where [latex]v[\/latex] is linear speed, [latex]s[\/latex] is displacement, and [latex]t[\/latex]
      \nis time.<\/p>\n

      [latex]v=\\frac{s}{t}[\/latex]<\/div>\n

      Angular speed<\/strong> results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as [latex]\\frac{360\\text{ degrees}}{4\\text{ seconds}}=[\/latex] 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where [latex]\\omega[\/latex] (read as omega) is angular speed, [latex]\\theta[\/latex] is the angle traversed, and [latex]t[\/latex] is time.<\/p>\n

      [latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/div>\n
      \n
      <\/div>\n
      \n
      \n

      angular speed<\/h3>\n

      As a point moves along a circle of radius [latex]r[\/latex], its angular speed<\/strong>, [latex]\\omega[\/latex], is the angular rotation [latex]\\theta[\/latex] per unit time, [latex]t[\/latex].<\/p>\n

      [latex]\\omega =\\frac{\\theta }{t}[\/latex]<\/p>\n<\/section>\n<\/div>\n<\/div>\n

      Combining the definition of angular speed with the arc length equation, [latex]s=r\\theta[\/latex], we can find a relationship between angular and linear speeds. The angular speed equation can be solved for [latex]\\theta[\/latex], giving [latex]\\theta =\\omega t[\/latex]. Substituting this into the arc length equation gives:<\/p>\n

      [latex]\\begin{align}s&=r\\theta \\\\ &=r\\omega t \\end{align}[\/latex]<\/div>\n

      Substituting this into the linear speed equation gives:<\/p>\n

      [latex]\\begin{align} v&=\\frac{s}{t} \\\\ &=\\frac{r\\omega t}{t} \\\\ &=r\\omega \\end{align}[\/latex]<\/div>\n
      \n
      \n

      linear speed<\/h3>\n

      The linear speed<\/strong>. [latex]v[\/latex], of the point can be found as the distance traveled, arc length [latex]s[\/latex], per unit time, [latex]t[\/latex].<\/p>\n

      [latex]v=\\frac{s}{t}[\/latex]<\/p>\n

      When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation<\/p>\n

      [latex]v=r\\omega[\/latex]<\/p>\n

      This equation states that the angular speed in radians, [latex]\\omega[\/latex], representing the amount of rotation occurring in a unit of time, can be multiplied by the radius [latex]r[\/latex] to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.<\/p>\n<\/section>\n

      How To: Given the amount of angle rotation and the time elapsed, calculate the angular speed.
      \n<\/strong><\/p>\n
        \n
      1. If necessary, convert the angle measure to radians.<\/li>\n
      2. Divide the angle in radians by the number of time units elapsed: [latex]\\omega =\\frac{\\theta }{t}[\/latex].<\/li>\n
      3. The resulting speed will be in radians per time unit.<\/li>\n<\/ol>\n<\/section>\n<\/div>\n
        A water wheel completes 1 rotation every 5 seconds. Find the angular speed in radians per second.\"Illustration<\/p>\n