This has come up a couple of times in the past week and, as every year, seems to be more troublesome than it ought to.
So – how should you model the forces acting on a moving lift?
Newton’s Second Law tells us that if there is an acceleration, there is also a force acting in the same direction, and this is the key to the problem. Remember these three ideas:
If the lift is accelerating upwards, there is an overall force upwards.
If the lift is accelerating downwards, there is an overall force downwards.
If the lift is moving at constant speed, there is no overall force because there is no acceleration.
So far, so good…. But when do you need the tension in the cable and when do you need the reaction on the floor?
The answer is that you use the tension when you are considering the lift and its contents, and the reaction on the floor when considering the contents alone.
For the whole lift we have:
where M is the mass of the lift and m is the mass of the contents.
If the lift is accelerating upwards the overall upward force is T – (Mg + mg).
If the lift is accelerating downwards the overall downward force is (Mg + mg) – T
I’ve swapped the expressions round to make the force positive. You can, of course, stick to one form or the other in which case the force and acceleration will come out negative when the lift is accelerating the opposite way.
And at constant speed, Mg + mg = T as there is no acceleration.
For the contents, we only consider forces acting directly on them – the weight and the reaction at the lift floor (R ).
If the lift is accelerating upwards the overall upward force is R – mg
If the lift is accelerating downwards the overall downward force is mg – R
And at constant speed, mg = R
Deceleration is simply negative acceleration, so a downward deceleration is the same as an upward acceleration, and vice versa.
Still confused? Then use the way your stomach feels in a moving lift.
As you set off going up or slow down while descending, there is an upward acceleration. You feel heavier – and your stomach tugs downwards.
As you set off going down or slow while ascending, there is an downward acceleration. You feel lighter – and your stomach tugs upwards.
And at constant speed, you feel your normal weight as you are not experiencing any extra acceleration.
Gut feel – sometimes it really does give you the right answer!