Simple Harmonic Motion in an Elevator

[Willjeezy]

Homework Statement

A simple pendulum is 5m long. What is the period of the oscillations for this pendulum in an elevator accelerating upwards at 5m/s² and accelerating downards at 5m/s²

Homework Equations

ω = √(g/L)

T = 2∏ / ω

The Attempt at a Solution

I got the right answers (trial and error), although I don't understand the concept.

for accelerating upwards.
ω = √[(9.8+5) /5]
ω = √[(14.8) /5]

^----this above part is confusing to me. If gravity is pointing down, and we have upward acceleration, shouldn't the value of (g + a) be smaller? I don't understand the meaning of 14.8 m/s²

I think the way the equation is defined confuses me.

ω = √(g/L)

the algebra of the equation doesn't confuse me, I understand why g can not be negative because of the root. However, if gravity always points down, and we let gravity be positive 9.8m/s, shouldn't and upward acceleration be defined as pointing in the negative direction?

thus shouldn't it be : ω = √[(9.8m/s² MINUS 5m/s²) /5] ??

can someone explain the sign conventions? I have confused myself.

 

[barryj]

Hmmm... I don't think you can always say that g points down. g is not a vector. The force that gravity causes is a vector and points toward the center of the Earth. The equation for the period of the pendulum is based on the force that pulls the bob downward. If the elevator is accelerating upward, then the tension on the string holding the bob will be greater. In the accelerating elevator, a person will feel like g is greater, right?

 

[Willjeezy]

you're analogy makes a lot of sense to me but I still don't really follow the math behind it.

What exactly is the g in that equation then if it is not a vector?

Essentially, I guess my question is if someone gave me that equation with the variables filled in and no prior information: ex:

ω = √[(9.8m/s² + 5m/s²) / 8] ??

And asked me to explain what all the numbers mean, I would be able to tell them that the 8 refers to the length.
9.8 is g

and I would have no clue what to tell them what +5m/s² means in terms of the equation. Can anyone help me make sense of this?

 

[barryj]

According to Newtons gravitational laws, thne force between any two masses is..

F = (G X M1 X M2)/r^2 where G in this case is the universal gravitational constant, M1 and M2 are the respective masses, and r is the distance between them.

When you recall that F = mg the g term is actually (G X M1)/r^2 where G again is the universal gravitational constant= 6.672E-11, M1 is the mass of the Earth = 5.98E24, and r is the radius of the Earth = 6.36E10. Work out these constanta and you will get g = 9.81 m/sec^2.

 

[Willjeezy]

hmm. Thanks for your replies barryj. I guess my problem is that I'm not 100% sure which equations require -9.8m/s and which use 9.8m/s.

 

[barryj]

Have you looked into the derivation of the equation ω = √(g/L) ? This might help. Actually the derivation shows a free body diagram with the string tension, the angle off vertical, and the force due to gravity (and acceleration). After some math manipulations, the ω = √(g/L) is derived. Notice that there is no mass involved.

 

[haruspex]

I don't think you can always say that g points down. g is not a vector.

g is an acceleration, and acceleration is a vector. But the pendulum is not being allowed to accelerate downwards at rate g, so it does not make sense to add to it, vectorially, an acceleration that is occurring. In fact, if you apply the equivalence principle, an object held up by a floor in a gravitational field strength g is equivalent to an object being accelerated upwards at g in the absence of a gravitational field.