{"id":9446,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9446"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"rotational-kinematics","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/rotational-kinematics\/","title":{"rendered":"Rotational Kinematics"},"content":{"rendered":"

Unit: <\/strong>Torque and Rotational Motion<\/strong><\/h2>\n

Chapter: <\/strong>Rotational Kinematics<\/strong><\/p>\n

Torque and Rotational Motion, Rotational Kinematics, <\/em>Kinematics of Rotational Motion About a Fixed Axis<\/em>, <\/em>Angular displacement, Angular velocity, Angular acceleration <\/em><\/h3>\n

After studying this chapter, you should be able to:<\/strong><\/p>\n

    \n
  • define the Kinematics of Rotational Motion About a Fixed Axis<\/li>\n
  • define angular displacement, angular velocity, angular acceleration  <\/li>\n<\/ul>\n

    Kinematics of Rotational Motion About a Fixed Axis<\/strong><\/p>\n

    The kinematical quantities in rotational motion, angular displacement (q), angular velocity (w) and angular acceleration (a) respectively are analogous to kinematic quantities in linear motion, displacement (x), velocity (v) and acceleration (a). We know the kinematical equations of linear motion with uniform (i.e., constant) acceleration:<\/p>\n

    where x0<\/sub> = initial displacement and v0<\/sub> = initial velocity. The word ‘initial’ refers to values of the quantities at t = 0<\/p>\n

    The corresponding kinematic equations for rotational motion with uniform angular acceleration.<\/p>\n

    Important Definitions<\/strong><\/p>\n

    Angular displacement (<\/strong>q<\/strong>): <\/strong><\/p>\n

      \n
    • Angular displacement is a measure of the change in the orientation of an object with respect to a reference point or axis. It is usually measured in radians or degrees and represents the angle through which an object has rotated or moved along a circular path.<\/li>\n
    • Angular displacement can be positive or negative, depending on the direction of rotation. If an object rotates counterclockwise, its angular displacement is positive, while if it rotates clockwise, its angular displacement is negative. The magnitude of the angular displacement is equal to the angle between the object's initial and final positions.<\/li>\n
    • Angular displacement is a fundamental concept in physics and is used to describe the motion of rotating objects such as gears, wheels, and planets. It is also used in many engineering and technological applications, such as robotics, computer graphics, and animation.<\/li>\n<\/ul>\n

      Angular velocity (<\/strong>w<\/strong>):<\/strong><\/p>\n

      Angular velocity is a measure of how fast an object is rotating around a fixed axis. It is the rate at which an object is changing its angular position over time and is typically measured in radians per second (rad\/s). The angular velocity is equal to the change in the angle of rotation divided by the time interval over which the angle changes. Mathematically, it can be represented as:<\/p>\n

      ω = Δθ\/Δt<\/strong><\/p>\n

      where ω is the angular velocity, Δθ is the change in angle of rotation, and Δt is the time interval over which the angle changes. The angular velocity vector is perpendicular to the plane of rotation, and its direction is determined by the right-hand rule.<\/p>\n

       <\/p>\n

      Angular acceleration (<\/strong>a<\/strong>):<\/strong><\/p>\n

      Angular acceleration is the rate at which an object's angular velocity changes with respect to time. It is a measure of how quickly the object is speeding up or slowing down as it rotates around a fixed axis. Angular acceleration is a vector quantity, and its direction is determined by the direction of the change in angular velocity.<\/p>\n

      The formula for angular acceleration is:<\/p>\n

      α = Δω \/ Δt<\/strong><\/p>\n

      where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the time interval over which the change occurs.<\/p>\n

      Angular acceleration is typically measured in units of radians per second squared (rad\/s2<\/sup>). It is an important concept in physics and engineering and is used to describe the behaviour of rotating objects such as wheels, gears, and turbines.<\/p>\n

      Example 1.<\/strong><\/p>\n

      The angular speed of a motor wheel is increased from 1200 rpm to<\/p>\n

      3120 rpm in 16 seconds. (i) What is its angular acceleration, assuming the acceleration to be uniform? (ii) How many revolutions does the engine make during this time?<\/p>\n

      Answer<\/strong><\/p>\n

      (i)   We shall use w = w0<\/sub> + at<\/p>\n

      w0<\/sub> = initial angular speed in rad\/s<\/p>\n

          = 2p × angular speed in rev\/s<\/p>\n

                  =<\/em>2π×angular speed in rev\/min60 s\/min<\/em><\/p>\n

                  =<\/em>2π×1200<\/em>60rad\/s<\/em><\/p>\n

         = 40p rad\/s<\/p>\n

      Similarly, w = final angular speed in rad\/s<\/p>\n

                               =<\/em>2π×3120<\/em>60rad\/s<\/em><\/p>\n

      = 2p × 52 rad\/s<\/p>\n

      = 104p rad\/s<\/p>\n

                   Angular acceleration<\/p>\n

                        The angular acceleration of the engine = 4p rad\/s2<\/sup><\/p>\n

      (ii) The angular displacement in time t is given by<\/p>\n

                    q = w0<\/sub>t + 1\/2<\/em> at2<\/sup><\/p>\n

                       = (40p × 16 + 1\/2<\/em> × 4p × 162<\/sup>) rad<\/p>\n

                       = (640p + 512p) rad<\/p>\n

      Key points:<\/strong><\/p>\n

      Rotational kinematics deals with the motion of objects that rotate or spin around an axis. The key quantities involved in rotational kinematics are:<\/p>\n

        \n
      • Angular displacement (θ): The change in the angle of a rotating object with respect to a fixed axis.<\/li>\n
      • Angular velocity (ω): The rate at which the object rotates, measured in radians per second.<\/li>\n
      • Angular acceleration (α): The rate at which the angular velocity of the object changes, measured in radians per second squared.<\/li>\n
      • Moment of inertia (I): A property of the object that describes its resistance to rotational motion.<\/li>\n
      • Torque (τ): The force that causes rotational motion, measured in newton meters (N·m).<\/li>\n
      • The relationships between these quantities are described by several important equations, including:<\/li>\n<\/ul>\n

         <\/p>\n

        θ = ωt, where θ is the angular displacement, ω is the angular velocity, and t is the time.<\/p>\n

        ω = ω0 <\/sub>+ αt, where ω0<\/sub> is the initial angular velocity.<\/p>\n

        θ = 1\/2 (ω0<\/sub> + ω)t, where t is the time.<\/p>\n

        ω2<\/sup> = ω0<\/sub>2<\/sup> + 2αθ, where θ is the angular displacement.<\/p>\n

        τ = Iα, where τ is the torque and I is the moment of inertia.<\/p>\n

         <\/p>\n

        These equations can be used to solve problems involving the motion of rotating objects, such as the motion of a spinning top or the rotation of a wheel.<\/p>\n

         <\/p>\n","protected":false},"excerpt":{"rendered":"

        Unit: Torque and Rotational Motion Chapter: Rotational Kinematics Torque and Rotational Motion, Rotational Kinematics, Kinematics of Rotational Motion About a Fixed Axis, Angular displacement, Angular velocity, Angular acceleration After studying this chapter, you should be able to: define the Kinematics of Rotational Motion About a Fixed Axis define angular displacement, angular velocity, angular acceleration   […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[622],"tags":[],"class_list":["post-9446","post","type-post","status-publish","format-standard","hentry","category-ap-physics-1"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9446","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9446"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9446\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9446"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9446"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}