The discussion revolves around calculating the average velocity and acceleration of the tip of a 2.4 cm long hour hand of a clock from noon to 6 pm. Participants are exploring the relationships between angular and linear velocities, as well as the implications of these calculations in a Cartesian coordinate system.
Several participants have provided guidance on the need to clarify the definitions of initial and final velocities. There is ongoing exploration of the relationship between angular and linear velocities, with some participants expressing uncertainty about the calculations and the unit vectors involved.
Participants are navigating through potential misunderstandings regarding the time intervals and the nature of the motion of the hour hand, as well as the mathematical relationships between angular displacement and linear velocity.
If the book didn't match my result in post #29, then the book must be wrong. Now for the direction of the unit vector. In what direction is the tip of the hour hand moving at noon (i or j)? In what direction is the tip of the hour hand moving at 6 pm (-i or -j)? (j is the up-down direction, and i is the right-left direction).negation said:I did an update prior to this current post. I got the right magnitude. However, I cannot make sense of the unit vector.
Chestermiller said:If the book didn't match my result in post #29, then the book must be wrong. Now for the direction of the unit vector. In what direction is the tip of the hour hand moving at noon (i or j)? In what direction is the tip of the hour hand moving at 6 pm (-i or -j)? (j is the up-down direction, and i is the right-left direction).
These are very good questions. The velocity is changing with time (its direction is changing), so the average velocity between the initial time and the final time is only an approximation to the instantaneous velocity at some point between the initial time and the final time. The time at which the average velocity best approximates the instantaneous velocity is at the half-way point, 3 pm. At 3 pm, the instantaneous velocity is -3.5E-6j, while the average velocity is -2.22E-6 j. So the average velocity gives the correct direction for the instantaneous velocity at the half way point, and it is only about 30% lower in magnitude. This is pretty good, considering the huge time interval and the huge change in the displacement vectors over this time interval.negation said:It struck me while I was in the toilet I made a blunder.
At 12 position: (3.5e-6,0)
Velocity is in the +i direction, j is irrelevant here since its magnitude is 0.
At 6 position: (-3.5e-6i,0)
Velocity is in the -i direction.
p(6) - p(12) = (-7e-6,0)/21600
what I got now is -3.24 e-10. This exactly coincides with the book.
Now, it would be helpful if you could shed some light on the conceptual difference between
v = -2.22e-6 in part(a) and v = rω = 3.5e-6
While I under v = rω implies linear velocity, I do not understand why it is not mathematically valid to use v = -2.22e-6 instead of 3.5e-6