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 — phy131studiof17:lectures:chapter10a [2017/08/04 13:41] (current) Line 1: Line 1: + ~~SLIDESHOW~~ + ====== Chapter 10 - Rotational Kinematics ====== + ===== Polar coordinates for rotational motion ===== + + {{angulardisplacement.png}} + + If we consider two points on a turning disc at distance $r_{1}$ and $r_{2}$ from the center of the disc we can make a number of observations. + + First the distance traveled by the points is quite different, points that are far from the center go through a greater distance. + + However, the angular displacement is the same for both points, suggesting that angular displacement is a good variable to describe the motion of the whole rotating object. + + We would like to be able to have a straight forward relationship between the actual distance traveled and the angular displacement,​ which we can have if we express our angular displacement in [[wp>​Radian|radians]]. + + As the relationship between the arc length on a circle and the angle which it subtends in radians is + + $l=r\theta$ + + a change in the angular displacement of $\Delta\theta$ results in a change in the displacement ​ + + $\Delta l=r\Delta\theta$ + + ===== Angular velocity===== + + For change in angular displacement $\Delta \theta$ in a time interval $\Delta t$ we can define an average angular velocity + + $\bar{\omega}=\frac{\Delta \theta}{\Delta t}$ + + As we are now accustomed, we can also define an instantaneous velocity + + $\omega=\lim_{\Delta t\to 0}\frac{\Delta \theta}{\Delta t}=\frac{d\theta}{dt}$ + + The units of angular velocity can be expressed as either $\mathrm{rad\,​s^{-1}}$ or $\mathrm{s^{-1}}$ + + ===== Angular acceleration ===== + + Previously we only considered circular motion in which the angular velocity $\omega$ remained constant with time, but of course we can also define an angular acceleration + + $\bar{\alpha}=\frac{\Delta \omega}{\Delta t}$ + + $\alpha=\lim_{\Delta t\to 0}\frac{\Delta \omega}{\Delta t}=\frac{d\omega}{dt}$ + + The units of angular acceleration can be expressed as either $\mathrm{rad\,​s^{-2}}$ or $\mathrm{s^{-2}}$ + + ===== From angular to tangential quantities ===== + + + A point at distance $r$ from the center of rotation will have at any time a tangential velocity of magnitude ​ + + $v=\omega r$ + + and a tangential acceleration (not to be confused with the centripetal acceleration) of + + $a=\alpha r$ + + ===== Useful relationships concerning the angular velocity ===== + + Rotational motion, when not accelerated,​ can be considered to be a form of periodic motion, and so relationships between the angular velocity, frequency and period are useful. + + $T=\frac{2\pi}{\omega}$ + + $f=\frac{\omega}{2\pi}$ + + $\omega=2\pi f$ + + We'd also like to be able to express the centripetal acceleration in terms of $\omega$ + + $a_{R}=\frac{v^{2}}{r}=\frac{(\omega r)^{2}}{r}=\omega^{2}r$ + + ===== About that acceleration.. ===== + + {{totalrotacc.png}} + + The total acceleration of an object in accelerated rotational motion will be the vector sum of two perpendicular vectors, the tangential acceleration $\vec{a}_{tang}$,​ and the radial acceleration $\vec{a}_{R}$. + + The magnitude of the total acceleration is + + $\vec{a}=\sqrt{a_{tang}^2+a_{R}^2}=\sqrt{\alpha^2r^2+\omega^{4}r^2}=r\sqrt{\alpha^2+\omega^4}$ + + and it is directed an angle + + $\arctan\frac{\alpha}{\omega^{2}}$ away from the radial direction. + + ===== Pseudovector representation of angular velocity and acceleration ===== + + As we know, translational velocity and acceleration are vector quantities. While we have defined angular velocity and acceleration we can see that they can represent points that are moving in directions that change with time, which would make them difficult to represent with vectors defined in cartesian coordinates. + + We can however consider an axial vector, ie. a vector that points in the direction about which rotation occurs as a good way of representing the direction of these rotational quantities. + + We give the direction of rotation around the axes according to a [[wp>​Right-hand_rule|right-hand rule]]. + + {{rhgriprule.png}} + + + + ===== Equations of motion for rotational motion ===== + + At the beginning of the course we derived, using calculus, a set of equations for motion under constant acceleration. + + $v= v_{0}+at$ ​ + + $x= x_{0}+v_{0}t+\frac{1}{2}at^2$ + + $v^{2}=v_{0}^2+2a(x-x_{0})$ + + We can equally derive similar equations for our rotational quantities. Indeed as we can see that the relationships between the new rotational quantities we have now are exactly the same as those between the translational quantities we can simply rewrite the translational motion equations in terms of rotational variables. + + $\omega= \omega_{0}+\alpha t$ + + $\theta= \theta_{0}+\omega_{0}t+\frac{1}{2}\alpha t^2$ + + $\omega^{2}=\omega_{0}^2+2\alpha(\theta-\theta_{0})$ + + ===== Combining translation motion with rotational motion - rolling ===== + /​*<​html>​ +
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