Hang a pendulum inside an accelerating train and the string comes to rest at an angle. Start by watching from outside the train, standing on the ground.
The bob is accelerating along with the train, so its forces are not in balance. A horizontal resultant of is required. Only two forces act on the bob, the tension and gravity , so the sum of those two must point horizontally and come to .
In components: horizontally , and vertically, where there is no acceleration, . Dividing one by the other removes and leaves .
The angle is fixed by the acceleration alone. Neither the mass of the bob nor the length of the string appears. And so far no inertial force has been used even once.
Now board the train and look at the same bob. It hangs at its angle, motionless in front of you. Since it is not moving, the forces on it ought to balance.
But the forces acting are the same two as before, the tension and gravity. Add them and a horizontal is left over, so they do not balance. It looks stationary and yet there is force to spare.
So we add one more force, of size , pointing opposite to the acceleration. Now three forces balance and the account agrees with the bob being at rest. This force is called an inertial force.
A person standing outside cannot see this force. An inertial force appears only once you have moved into the accelerating frame, purely to make the books balance. Search for something exerting it and you will not find it.
Gravity and the inertial force have a great deal in common. Both have a size proportional to the mass of the bob, and neither changes in size or direction. So the two may be gathered into a single force.
Together their size is , and their direction is tilted from the vertical by . Regarded as one gravity, this says that inside the train the acceleration due to gravity is and that its direction is down. This is called the apparent gravity.
The pendulum hangs along this tilted down and rests there. Displace it and it swings about that direction. The dashed line in the figure is the vertical inside this train.
The manner of swinging is unchanged, so the equation for the period carries over. Replace the in the previous lecture's with and it becomes .
is always larger than , so a pendulum in an accelerating train swings faster than one in a train at rest. The two in the figure are released together and drift steadily apart.
A lift is simpler still. The acceleration is vertical, so down does not tilt and only the size changes, to . Accelerating upward shortens the period, accelerating downward lengthens it. In free fall and the pendulum no longer swings at all.