- #1
PFuser1232
- 479
- 20
In the book "Introduction to Mechanics" by K&K, an increment of a generic time-varying vector is split into two components, ##\Delta \vec{A} _{\perp}## and ##\Delta \vec{A}_{\parallel}##.
Their magnitudes are approximated by:
$$A \Delta \theta$$
and
$$\Delta A$$
respectively. (Where ##\Delta \theta## is the angle between ##\vec{A}(t)## and ##\vec{A}(t + \Delta t)##)
Now, I understand why the parallel component of the increment is approximated by ##\Delta A##, it's because:
$$|\Delta \vec{A}_{\parallel}| = |\vec{A}(t + \Delta t)| cos(\Delta \theta) - |\vec{A}(t)| ≈ |\vec{A}(t + \Delta t)| - |\vec{A}(t)| = \Delta A$$
for small angles. However, I can't understand how the other approximation is obtained (when changes in magnitude and direction are occurring simultaneously, that is).
My second question:
To find the magnitudes of the derivatives of the vector components, we divide each of the expressions I mentioned above by ##\Delta t##, and we take the limit as ##\Delta t## approaches zero. Why did we have to approximate the magnitudes (using the small angle approximation) if we were going to take the limit as ##\Delta t## approaches 0 anyway? Would it make a difference if we were to use the actual equations with strict equality that hold for any angle, big or small, and then take the limit as ##\Delta t## approaches zero?
Their magnitudes are approximated by:
$$A \Delta \theta$$
and
$$\Delta A$$
respectively. (Where ##\Delta \theta## is the angle between ##\vec{A}(t)## and ##\vec{A}(t + \Delta t)##)
Now, I understand why the parallel component of the increment is approximated by ##\Delta A##, it's because:
$$|\Delta \vec{A}_{\parallel}| = |\vec{A}(t + \Delta t)| cos(\Delta \theta) - |\vec{A}(t)| ≈ |\vec{A}(t + \Delta t)| - |\vec{A}(t)| = \Delta A$$
for small angles. However, I can't understand how the other approximation is obtained (when changes in magnitude and direction are occurring simultaneously, that is).
My second question:
To find the magnitudes of the derivatives of the vector components, we divide each of the expressions I mentioned above by ##\Delta t##, and we take the limit as ##\Delta t## approaches zero. Why did we have to approximate the magnitudes (using the small angle approximation) if we were going to take the limit as ##\Delta t## approaches 0 anyway? Would it make a difference if we were to use the actual equations with strict equality that hold for any angle, big or small, and then take the limit as ##\Delta t## approaches zero?