A simple pendulum is suspended in the lift and it's time period is T. What will happen to it's T when lift moves upwards or downwards?
Answers
Answer:
A simple pendulum suspended from the ceiling of a stationary lift has period T0. When the lift descends at steady speed, the period is T1, and when it descends with constant downward acceleration, the period is T2.
Answer:
. That is, accelerating the pendulum will change the period of oscillation in a predictable way. And there are several ways to look at the problem which come up in the various answers.
Things oscillate if there is always a restoring force that always points toward an equilibrium position. So when the pendulum is to the right of equilibrium, the force brings it back to the left. When it is to the left, the restoring force is to the right.
We probably should consider the pendulum as a rotation problem, but it is easier in this case to look at it assuming the angle the string makes with the vertical is always so small we can think of the mass itself just moving horizontally. In that case, the restoring force on the mass is just the horizontal component of the tension in the string - as in the image on the left (although it is shown with an exaggerated angle in order to show the components more clearly).
If the pendulum is now accelerated upward - as in the image on the right, the tension in the string increases, so the vertical component of the tension becomes greater than the gravitational force on the mass (in order to accelerate it upward) and hence the horizontal component of the tension also increases. But that component is the restoring force, so the mass moves toward equilibrium more quickly than when the pendulum was not being accelerated. That is, the frequency of oscillation increases, so the period decreases.
If the pendulum was not being held up - that is, if it were accelerating downward, that would reduce the tension in the string, mg would then be greater than the vertical component of the tension, and the horizontal component (that is, the restoring force) would also be reduced and it would take longer for the mass to return through equilibrium. That is, the period would decrease.
That last part is interesting if you think of the pendulum being in free fall. That is, if the top of the pendulum and the mass were both accelerating downward at acceleration g, then there would be no tension in the string, hence no restoring force - so the pendulum would not oscillate at all. And that is, in effect, what both Peter Upton and Steve Baker said in their answers - the equivalent of letting “g” go to zero in the equation for the period