- Statement of Problem:
- Consider a simple pendulum (mass m and length L)
- mounted inside a railroad car that is accelerating to the
- right with constant acceleration A a shown. Find the angle
- eq at which the pendulum will remain at rest relative to the accelerating car, and find the frequency of small oscillations about this equilibrium angle.
- Solution:
- This time we will use Newtonian mechanics and think about the forces on the plumb bob. There are two forces, mg down and the tension force T, i.e. F = T + mg. In the accelerating frame, as we saw, we need to write:
- where geff = g – A is a vector quantity that can be considered an effective gravity force. (This is why you feel heavier when taking off in an airplane, for example.)
- Thus, the equation of motion is just like for a normal pendulum so long as we replace g with geff.
- If the pendulum remains at rest, then The equil. angle is
Example 9.1, cont’d - Solution, cont’d:
- As you know, the frequency of oscillation of a pendulum is
- Again, we just replace g with
- You have done this problem using Lagrangian mechanics in the frame of the ground (Prob. 7.30), and you may have found it to be considerably more difficult that way. Working in the accelerating frame can be much easier, but recall that we cannot use Lagrangian mechanics directly in a non-inertial frame.
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