Easyphysics

Thursday 14 July 2011

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Wednesday 25 May 2011

Angular Acceleration

When we switch on an electric fan, we notice that its angular velocity goes on increasing. We say that it has an angular acceleration ( change in velocity called acceleration ). Angular acceleration can be defined as "the rate of change of angular velocity". If ω_i_andω_f_{are the values of instantaneous velocity of a rotating body at instants}t_i_andt_f_{, the average angular acceleration during the interval}t_f-t_i_{is given by}

$\fn_jvn \alpha _{av}=\frac{\omega _{f-}\omega _{i}}{t_{f-}t_{i}}=\frac{\Delta \omega }{\Delta t}$

The instantaneous angular acceleration is the limit of the ratio $\dpi{100} \fn_jvn \frac{\Delta \omega }{\Delta t}$ as $\dpi{100} \fn_jvn {\Delta t}$ approaches zero. Therefore, instantaneous angular acceleration is given by

$\dpi{100} \fn_jvn \alpha =\lim_{\Delta t\rightarrow 0}\frac{\Delta \omega }{\Delta t}$

The angular acceleration is also a vector quantity whose magnitude is given by the above equation and whose direction is along the axis of rotation, given by right hand rule. Angular acceleration is expressed in units of $\dpi{80} \fn_jvn rad$ / $\dpi{80} \fn_jvn sec^{2}$ .

Angular Velocity

Very often we are interested in knowing how fast or how slow a body is rotating. It is determined by its angular velocity which is defined as "the rate at which the angular displacement is changing with time". Referring to figure given below, if ∆Ѳ is the angular displacement during the time interval ∆t, the average velocity ω_av during this interval is given by,

$\fn_jvn \omega _{av}=\frac{\Delta \theta}{\Delta t}$

The instantaneous angular velocity ω is the limit of the ratio ∆Ѳ/∆t as ∆t, following instant t, approaches to zero.

$\fn_jvn \omega =\lim_{\Delta t \to 0}\: \frac{\Delta \theta }{\Delta t}$

In limit when ∆t approaches zero, the angular displacement would be infinitesimally small, so it would be a vector quantity and the angular velocity as defined by the above equation would also be a vector. Its direction is along the axis of rotation and is given by right hand rule.

Angular velocity (ω) is measured in radians per second which is SI unit. Sometimes it is also given in terms of revolution per minute.

Units of Angular Displacement

Three units are generally used to express angular displacement, namely degrees, revolution and radian. We are familiar with the first two. As regards radian which is SI unit, consider an arc of length S of a circle of radius r which subtends an angle Ѳ at that center of the circle given in radians (rad) as shown in the figure given below: