The rotation of a particle about an axis is specified by 2 pieces of information

Transcription

1 1 How to specify rotational motion The rotation of a particle about an axis is specified by 2 pieces of information 1) The direction of the axis of rotation 2) A magnitude of how fast the particle is "going round" the axis of rotation. Question : What mathematical object has both direction and magnitude? Answer :

2 Hence we may specify rotational motion by a single angular velocity vector such that a) The magnitude = angular speed b) points in the direction of the axis of rotation Question : Is this enough? Answer :

3 2. Right - hand convention 1) Take the right hand. 2) Wrap fingers in the direction of rotation. 3) Thumb points in direction of the angular velocity vector. Exercise : Which way does point?

4 3. Rotational motion in 3-d Question : In circular motion v = R. How do we generalize to 3-d? Z A R O r(t) Y P X

5 Z A R P O r(t) Y 1) R = AP = r sin X 2) speed of P, v = R = r sin 3) v is perpendicular to r and

6 Hence (using right hand convention) for rotational motion v = dr / dt = x r This is the generalization of the formula for circular motion

7 Common misconceptions 1. 3d - all quantities are vectors! v = x r 2.. Vector cross product not scalar multiplication! a) Order is important b) v x r 3. Position vector not radius of circle! Advice : Purge scalar formula v = r from memory

8 Exercise : For circular motion in the XY plane the position vector r(t) = {R cos t, R sin t, 0} Y a) Find the angular velocity vector b) Using find the velocity vector v X a) = b) v = x r Note : Compare with dr/dt

9 4 Rotating axes system Definition An axes system oxyz is said to be rotating with respect to a fixed axes system OXYZ if Z 1) The origin of both axes systems coincide 2) The orientation of the ox, oy and oz axes are all changing with the same angular velocity vector. x O = o z y Y Question : What happens if they don t? X

10 Implications 1) Since the orientation of the oxyz changes cannot ignore di/dt, dj/dt and dk/dt 2) What s di/dt, dj/dt and dk/dt? Hint : dr/dt = v = x r for rotation Hence di / dt = dj / dt = dk / dt = x i x j x k

11 5 Differentiating a vector expressed in terms of a rotating frame Given basis vectors of rotating frame i, j, k position vector of a particle P, r(t) = x(t) i + y(t) j + z(t) k angular velocity vector of the rotating frame (t) = x (t) i + y (t) j + z (t) k Show that the absolute velocity of P is dr/dt = x ' (t) i + y ' (t) j + z ' (t) k + x r Question : Does this make sense? What happens if is 0?

12 1. Position vector r(t) = x(t) i + y(t) j + z(t) k 2. Absolute velocity v = dr(t) / dt = dx/dt i + x di/dt + dy/dt j + y dj/dt + dz/dt k + z dk/dt 3. Use di/dt = x i dj/dt = x j dk/dt = x k

13 Substituting... v = dx/dt i + x ( x i) + dy/dt j + y ( x j) + dz/dt k + z ( x k) 4) Regroup terms v = dx/dt i + dy/dt j + + dz/dt k + x (xi + y j + z k) 5. Hence v = r + x r

14 6 How to find absolute acceleration for a rotating frame Denote r ' (t) = x ' (t) i + y ' (t) j + z ' (t) k r '' (t) = x '' (t) i + y '' (t) j + z '' (t) k where i, j, k are the basis vectors of a rotating frame Show that the absolute acceleration d 2 r/dt 2 = r '' + ' x r + x ( x r ) + 2 x r '

15 v = dr/dt = r + x r a = dv/dt = v + x v = (r ' + ( x r ) ) + x (r ' + ( x r ) ) = r '' + ' x r + 2 x r ' + x ( x r )

16 Exercise : Circular motion using rotating axes The particle P rotates about the OZ axis with angular speed (t) rad/s. OP = R m. The axes i j k rotates with the particle. a) Find the absolute velocity and acceleration of P using the rotating axes axes i j k. J b) Compare the derivation with j the one using the (t) rad/s fixed axes I J K. i P O = o R I Fixed

17 J j (t) rad/s i rotating R O = o I Fixed 1. Position vector r = 2. Angular velocity = 3. Absolute velocity v = dr/dt = r + x r =

18 4. Absolute acceleration a = dv/dt = v + x v = = Identify the terms