Classical Mechanics. SOLUTIONS by Gregory

By Gregory

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In still air, the aircraft can cruise with speed v A in any direction. When a steady wind is blowing, this remains true when the aircraft is observed from a frame moving with the wind. 4. The speeds v A and v W (and the angle ˛) are given, and we wish to choose the angle ˇ so that the velocity v G points north. Let the unit vectors fi ; j g be as shown, with i pointing east and j pointing north. Then, on taking components of equation (1) in the i - and c Cambridge University Press, 2006 56 Chapter 2 Velocity, acceleration and scalar angular velocity j -directions, we obtain 0 D v A sin ˇ C v W sin ˛; v G D v A cos ˇ v W cos ˛: The first equation shows that the aircraft heading ˇ is sin ˇ D  vW vA  sin ˛; and the second equation then determines the ground speed v G .

B2 b1 /. Then the rolling condition at the point of contact of G and the Sun gear G1 gives ! b2 2 b1 /: The rolling condition at the point of contact of G and the ring gear G2 gives ! 2 b2 D v C ! D ! 2 b2 b2 ! b1 C b2 / and the angular velocity  of the arm satisfies the equation L D v. Hence D v ! 1 b1 C ! 2 b2 D : L b1 C b2 c Cambridge University Press, 2006 53 Chapter 2 Velocity, acceleration and scalar angular velocity Problem 2 . 14 shows a straight rigid link of length a whose ends contain pins P , Q that are constrained to move along the axes OX , OY .

Find the angular velocity ! and the speed of the centre C of the link at time t . Solution Let  be the angle between the rod and the negative x-axis. 14) is ! D . a sin  / . D D xP a sin  b cos t b cos t D 1=2 a sin  a2 a2 cos2  b cos t D a2 2 2 b sin t 1=2 : This is the angular velocity of the rod at time t . X; Y /. Then X D 12 a cos ; Y D 21 a sin ; and so XP D YP D Hence XP 2 C YP 2 D 1 4 1 a sin  2 1 a cos  2   P ; P :   a2 sin2  P 2 C D 41 a2 P 2 D   1 4   a2 cos2  P 2 2 a2 b 2 cos2 t  : 4 a2 b 2 sin2 t c Cambridge University Press, 2006 Chapter 2 Velocity, acceleration and scalar angular velocity The speed of C at time t is therefore abj cos t j  1=2 : 2 a2 b 2 sin2 t c Cambridge University Press, 2006 54 55 Chapter 2 Velocity, acceleration and scalar angular velocity Problem 2 .

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