\n Circular motion is a two-dimensional motion in which an object moves along the circumference of a circular path.\r\n In circular motion, the velocity of the object keeps changing even if it is moving with a constant speed since the\r\n direction of motion changes at every point. Therefore, circular motion is always an accelerated motion.\n
\r\n\n In circular motion, kinematics is studied with the help of two kinds of variables — linear variables and angular\r\n variables. Let's have a look at each one of them one by one.\n
\r\n\n Angular velocity is the angle in radians that a body moving in circular motion rotates per unit time. In other\r\n words, it is the rate of change of the angle \\(\\theta\\) moved by the body. It is denoted by \\(\\omega\\) and is measured in rad/s.\r\n If the body rotates by an \\(\\Delta \\theta\\) in time \\(\\Delta t\\), angular velocity is given as:\r\n $$\\text{Angular velocity}=\\omega=\\lim_{\\Delta t \\to \\theta} \\frac{\\Delta \\theta}{\\Delta t}=\\frac{\\Delta \\theta}{dt}$$\r\n
\r\n\r\n Linear velocity of a body moving in circular motion is given by:\r\n $$\\text{Linear velocity}=\\mathbf{v}=\\frac{d\\mathbf{s}}{dt}$$\r\n where \\(\\mathbf{s}\\) is the displacement of the body.\r\n
\r\n The magnitude of linear velocity is called linear speed and is given as $$v=\\left|\\frac{d\\mathbf{s}}{dt}\\right|$$\r\n Linear speed and angular speed are related as follows: $$v=r\\omega$$\r\n
\r\n It is the rate of change of angular velocity of the body. It is denoted by α.\r\n $$\\text{Angular Acceleration}=\\alpha=\\frac{d\\omega}{dt}=\\frac{d^2\\theta}{dt^2}$$\r\n
\r\nIn circular motion, linear acceleration is divided into two components:
\r\n\n Net acceleration of the body is given as the vector sum of the two components of acceleration given\r\n above.\r\n $$\\text{Net acceleration}=a=\\sqrt{a_t^2+a_r^2}$$\r\n
\r\n\r\n\n If the object moving in circular motion moves with a constant speed, its motion is called uniform circular motion.\r\n Its tangential acceleration is zero but centripetal acceleration is not zero.\n
\r\n\n In circular motion, the direction of the particle changes at every moment. At each moment instead of travelling\r\n straight, it turns towards the center, therefore there must be a force responsible for changing the direction of\r\n the particle. This force responsible for turning the direction of motion of the body towards the center of the\r\n circle is called the centripetal force. It is always directed towards the center of the circle.\r\n
\r\n\n For example, when moon moves around the earth in a circular path, the gravitational force of attraction on the\r\n moon by the earth provides the required centripetal force.\r\n Now let us have a look at the magnitude of centripetal force.\r\n $$\r\n \\begin{aligned}\r\n \\text{Centripetal Force} &= m \\times a_r \\\\\r\n &= m \\times \\frac{v^2}{r} \\\\\r\n &= \\frac{mv^2}{r} \\\\\r\n &= m \\omega^2 r \\quad (\\text{Since } v=r\\omega)\r\n \\end{aligned}\r\n $$\r\n where \\(v\\) is the linear speed, \\(\\omega\\) is the angular speed and \\(r\\) is the radius of the circular path.\r\n
\r\n\n Centrifugal force is not an actual force, it is the pseudo force that appears to act on the object moving in a\r\n circular motion when viewed in a rotating frame of reference. It is directed away from the center of the circular\r\n path the object is moving in. For a body of mass m rotating in circular path of radius \\(r\\) with an angular velocity\r\n \\(\\omega\\), magnitude of centrifugal force is equal to\r\n \\(m \\omega^2 r\\) or \\(m v^2 /r\\).\r\n $$F = \\frac{mv^2}{r} = m \\omega^2 r$$\r\n\r\n
Let’s take an example. Suppose a stone is tied to a string and it is whirled around the string. When it is viewed\r\n from a non-inertial frame of reference (rotating frame of reference) it appears to be stationary however, we know that\r\n the force applied by the string is still acting on the stone towards the center therefore a pseudo force is applied\r\n in a direction away from the center which is the centrifugal force in this case.\n
\n Shown is a carnival ride that travels along a circular path with a constant linear speed \\(v_0\\).\r\n As the ride travels along the circular path, the tension in the bar connecting the center to the ride changes.\r\n Through this MagicGraph, students will learn:\n
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