Phys 325 Spring 2011 Lecture 22 - Physics 325 Lecture 22...

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Physics 325 Lecture 22 Mechanics in non-inertial reference frames In discussing Newton’s Laws, we made clear that they are only valid in the special case of inertial reference frames – frames that are neither accelerating nor rotating. It is often desirable to consider non-inertial frames, however. An important example is the frame fixed to the Earth. This frame is to very good approximation inertial, but a precise treatment of mechanics using an Earth-fixed frame (e.g. description of long-range rocket motion) requires consideration of the Earth’s rotation about its axis and acceleration in its orbit around the Sun. With the goal of understanding the mechanics of bodies in non-inertial frames, we’ll start by considering the simple case of acceleration without rotation. Then we’ll discuss rotating reference frames. Uniform acceleration without rotation We consider the motion of a particle whose coordinates are referenced to a non-inertial reference frame which is accelerating with respect to ―fixed‖ or inertial axes. We use i x (primed) as coordinates in the fixed system and i x (unprimed) as coordinates in the accelerating system. We have r R r  where R locates the origin of the accelerating system in the fixed system. Doubly differentiating with respect to time gives r R r which we will write as r r R  Multiplying through by m and using the fact that F mr since the primed frame is inertial, we have mr F mA (22.1) where AR .
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Equation (22.1) has exactly the form of Newton’s 2 nd Law, except that in addition to the net force in the inertial frame F , there is an extra term on the right equal to mA . This means that we can continue to use Newton’s 2 nd Law in the non-inertial frame provided we agree that we must add an extra force-like term, often called the inertia force inertial F given by inertial F mA  (22.2) Example: Pendulum in an accelerating car Lets work out the angle of a pendulum that hangs plumb with respect to a car with acceleration A . Lets first do this the ―traditional way‖ using Newton’s 2 nd Law and only referencing coordinates to an inertial frame (e.g. the road). As observed in any inertial frame, there are only two forces acting on the bob: the tension in the string T and the weight mg . Therefore, we have F T mg mA (22.3) and sin cos 0 T mA T mg  Combining the two component equations gives tan A g (22.4) If we choose to do the problem in the non-inertial frame of the accelerating car, then we have a statics problem ( 0 F ) as shown in the figure below
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but we need to add the inertial force given by Equation (22.2) to give 0 F mr T mg mA   (22.5) This is equivalent to Equation (22.3) and therefore leads to same relation (Equation (22.4)). You can see that the inertial force is not a real force but rather a force-like term required to preserve the form of Newton’s 2 nd Law for a non-inertial reference frame. For this reason, inertial forces such the one shown in Equation (22.2) are often referred to as
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Phys 325 Spring 2011 Lecture 22 - Physics 325 Lecture 22...

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