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ONLINE: MATHEMATICS EXTENSION 2 Topic 6 6.61 MECHANICS MOTION IN A CIRCLE COMPLEX NUMBERS We can use complex numbers to mathematically investigate the motion of a particle in a circle. Let the X axis be the real axis and the Y axis be the imaginary axis. The position of the particle is given by the complex number z as a vector z x i y R cos i sin R ei where the angular displacement is a function of time t . The magnitude of z is z R and the angle of the complex vector z makes with the real axis is the argument of z Arg z physics.usyd.edu.au/teach_res/math/math.htm mec661 1 For motion in a circle the radius is constant dR d 2 R 2 0 dt dt RR0 The velocity v of the particle is found by differentiating the position of the particle z with respect to time t. dz z dt d d i i v R ei R i e i R e dt dt vi z v where the angular speed is d . dt The magnitude of the velocity is v R . Multiplication by i produces a rotation of a complex vector by /2 rad. Therefore z and i z are at right angles to each other. Hence, the displacement vector z and velocity vector v are always perpendicular to each other. physics.usyd.edu.au/teach_res/math/math.htm mec661 2 The acceleration is found by differentiating the velocity with respect to time dv v dt d d d i d i a i z i R ei i R e i e dt dt dt dt d ai R ei 2 R ei dt d d 2 d 2 i /2 2 a e i / 2 z z e z z 2 dt dt dt a where the angular acceleration is The first term e i /2 d . dt ddt z corresponds to the change in speed of the particle and the direction is perpendicular to the displacement vector. This term is the tangential acceleration aT. The magnitude of the tangential acceleration is d dv d aT e i / 2 z R dt dt dt The second term 2 z corresponds to the change in direction and it is in a direction opposite to the displacement vector. This is the radial acceleration aR. The magnitude of the radial acceleration is aR 2 z 2 R v2 R physics.usyd.edu.au/teach_res/math/math.htm mec661 3