16.07 Dynamics Fall 2004 Version 1.2 Lecture d13- Newton's Second law for non-Inertial observers Inertial forces Inertial reference frames In the previous lecture, we derived an expression that related the accelerations observed using two reference frames, A and B, which are in relative motion with respect to each other aB+(aM/B)ry+29×(vA/B)xy2x2+gxTA/B+92×(xr4/B) Here, aA is the acceleration of particle A observed by one observer, and(aa/B)xys, is the acceleration of the same particle observed thethe other (moving) observer. It is clear that the acceleration of particle A will be different for each observer, unless all the other terms in the above expression are zero. This means that if one of the observers is inertial, the other observer will be inertial if and only if n=0, S=0 and 0. Thus, we conclude tha inertial frames can not rotate with respect to each other, i.e., n=0 and S=0, and inertial frames can not be accelerating with respect to each other, i. e. aB=0 Thus, inertial frames can only be at most in constant relative velocity with respect to each other. In practical terms, the closest that we are able to get to an inertial frame is one which is in constant relative velocity with respect to the most distant stars Note The earth as an inertial reference frame Given that the earth is rotating about d at the same time is rotating about the sun. it is clear that the earth can not be an inertial reference frame. However, we shall see that, for many applications, the error made in assuming that the earth is an inertial reference frame is small We can easily estimate the effect of earth's rotation. Consider for instance two reference frames yz and x'yz. The first frame is fixed and the second frame rotates with the earthJ. Peraire 16.07 Dynamics Fall 2004 Version 1.2 Lecture D13 - Newton’s Second Law for Non-Inertial Observers. Inertial Forces Inertial reference frames In the previous lecture, we derived an expression that related the accelerations observed using two reference frames, A and B, which are in relative motion with respect to each other. aA = aB + (aA/B)x′y′z ′ + 2Ω × (vA/B)x′y′z ′ + Ω˙ × rA/B + Ω × (Ω × rA/B) . (1) Here, aA is the acceleration of particle A observed by one observer, and (aA/B)x′y′z ′ is the acceleration of the same particle observed the the other (moving) observer. It is clear that the acceleration of particle A will be different for each observer, unless all the other terms in the above expression are zero. This means that if one of the observers is inertial, the other observer will be inertial if and only if Ω˙ = 0, Ω = 0 and aB = 0. Thus, we conclude that, • inertial frames can not rotate with respect to each other, i.e., Ω = 0 and Ω˙ = 0, and, • inertial frames can not be accelerating with respect to each other, i.e. aB = 0. Thus, inertial frames can only be at most in constant relative velocity with respect to each other. In practical terms, the closest that we are able to get to an inertial frame is one which is in constant relative velocity with respect to the most distant stars. Note The earth as an inertial reference frame Given that the earth is rotating about itself and at the same time is rotating about the sun, it is clear that the earth can not be an inertial reference frame. However, we shall see that, for many applications, the error made in assuming that the earth is an inertial reference frame is small. We can easily estimate the effect of earth’s rotation. Consider for instance two reference frames xyz and x ′y ′ z ′ . The first frame is fixed and the second frame rotates with the earth. 1