\n It is the rate of change of angular position of the particle with respect to time. It tells us\r\n how fast an object is rotating. It is a vector quantity and is denoted by \\(\\omega\\). It SI unit is rad/sec.\r\n If a body rotates by Δ\\(\\theta\\) angle in a time duration of Δ\\(t\\), its average angular velocity over the time period Δ\\(t\\) is given by\r\n $$\\omega=\\frac{\\Delta \\theta}{\\Delta t}$$\r\n When Δt tends to zero, we get the instantaneous angular velocity of the body which is given by\r\n $$\\omega=\\lim_{Δt\\to0}\\frac{Δθ}{Δt}=\\frac{dθ}{dt}$$\r\n Its direction is along the axis of rotation of the body, perpendicular to the plane of rotation. Direction of\r\n the angular velocity \\(\\omega\\) is given using the right-hand rule. If the fingers of the right hand are\r\n curled in the direction of rotation, the direction in which the thumb of the right hand points out, is the\r\n direction of angular velocity. Relation between angular velocity \\(\\vec{\\omega}\\) and linear velocity \\(\\vec{v}\\) of a particle or a body in vector\r\n form is given as,\r\n $$\\vec{v}=\\vec{\\omega} \\times \\vec{r}$$\r\n where \\(\\vec{r}\\) is the position vector of the particle.\r\n Linear velocity is the vector product of angular velocity and position vector of the particle in\r\n angular motion. However, in most of the cases the relation between angular velocity is given only in terms of\r\n their magnitude i.e., \\(v=\\omega r \\).\r\n Note that for rotation of a body about a fixed axis, direction of angular velocity remains constant however, its\r\n magnitude may change.\r\n
\r\n\n Angular acceleration is the rate of change of angular velocity. It is denoted by \\(\\alpha\\) and its SI unit is rad/s2. It is\r\n also a vector quantity and its direction is also directed perpendicular to the plane of rotation. We have\r\n read how to find the direction of angular velocity. Now, if the angular velocity of the particle or object\r\n in rotational motion is increasing, the direction of angular acceleration is same as that of angular\r\n velocity and if the angular velocity is decreasing, the direction of angular acceleration is in a direction\r\n opposite to the direction of angular velocity.\r\n Average angular acceleration is given by\r\n $$\\alpha= \\frac{Δ\\omega}{Δt}$$\r\n where Δt is the change in time and Δ\\(\\omega\\) is the change in angular velocity.\r\n
Instantaneous angular\r\n acceleration is given by\r\n $$\\alpha=\\lim_{Δt\\to0}\\frac{Δ\\omega}{Δt}=\\frac{d\\omega}{dt}$$\r\n
\n Moment of inertia of a body is a quantity that determines the body's resistance to change in its rotational\r\n motion. The role of moment of inertia in rotational motion is same as that of mass in linear motion. It\r\n determines the torque required for a specific angular acceleration about the axis of rotation.\r\n Moment of inertia depends\r\n upon the distribution of mass in the body about its rotational axis. The farther are the masses from the\r\n axis of rotation, more is the moment of inertia of the body.\n
\r\n\n For a point mass, moment of inertia is given by\r\n $$I =mr^2$$\r\n where \\(m\\) is the mass of the particle, \\(I\\) is its moment of inertia and \\(r\\) is the perpendicular\r\n distance of the\r\n particle from the axis of rotation.\n
\r\n\r\n For a system with n particles having masses\r\n \\(m_1, m_2, ... ,m_n\\), moment of inertia I of the system is given by\r\n $$I=\\sum_{i=1}^{n} m_ir_i^{2}$$\r\n Where \\(r_i\\) is respective perpendicular distance of particle of mass \\(m_i\\) from the axis of rotation.\r\n
\r\n\r\n For extended objects with continuous mass distribution, we consider infinitesimally small elements of mass\r\n \\(dm\\) and the summation in above formula is replaced by integration. Moment of inertia \\(I\\) of the system is given\r\n by\r\n $$I=\\int r^{2}\\,dm$$\r\n
\r\n\n Torque is the rotational analogue of linear force. It is responsible for rotating a body and bringing\r\n about an angular acceleration in the body.\r\n If a force \\(F \\) acts on a particle whose position vector is \\(\\vec{r}\\) with respect to the axis or point\r\n of rotation , then the torque \\(\\tau\\) acting on the particle is equal to the vector product of the force\r\n acting on the particle and its position vector i.e.,\r\n $$\\vec{\\tau}=\\vec{r} \\times \\vec{F}$$\r\n Here \\( \\tau \\) is the torque of the force applied. Torque is a vector quantity and its direction is\r\n determined according to the right-hand rule in a plane perpendicular to the force applied and position\r\n vector of the particle. Torque is denoted by \\(\\tau\\) and its SI unit is Newton-meter.\r\n
\r\n\n The magnitude of torque is given by\r\n $$\\tau=r F \\sin \\theta $$\r\n where \\(r\\) is the magnitude of position vector of the particle, \\(F\\) is the magnitude of the force applied\r\n and \\(\\theta\\) is the angle\r\n between the direction of the force applied and position vector of the particle.\r\n Therefore, the turning effect on an object due to a force applied not only depends on the magnitude and\r\n direction of the force applied but also on the point of application of the applied force.\r\n\r\n Angular acceleration produced in a body not only depends on the torque applied on it, but also\r\n on its moment of inertia. The relation between torque \\(\\tau\\), angular acceleration\\(\\alpha\\) and moment\r\n of inertia \\(I\\) of the body is given by\r\n $$\\vec{\\tau}=I\\vec{\\alpha}$$\r\n
\r\n\r\n Shown is a dvd disc that can spin about point 'O'.\r\n The disc has a mass \\(M_{disc}\\) and radius \\(r_{disc}\\).\r\n You will also see a ball (treated as a point mass) that has a mass \\(m_{ball}\\).\r\n Start with giving the disc an initial spin. The initial angular velocity of the disc is \\(\\omega_i\\) — which can be changed.\r\n Then place the ball on the rim of the disc, and observe the new angular velocity of the combined system (\\(\\omega_f\\)).\r\n Change the mass of the ball and note how angular velocity changes.\n
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