Where Did I Go Wrong in Calculating the Pendulum Period on an Accelerating Jet?

[Elfrae]

"As your jet plane speeds down the runway on take off, you measure its acceleration by suspending your yo-yo as a simple pendulum and noting that when the bob (mass 40g) is at rest relative to you, the string (length 70cm) makes an angle of 22 degrees with the vertical. Find the period T for small angle oscillations of this pendulum."

I've tried to do this using total force on the bob = tension + mg + ma, where a is the acceleration of the plane. Resolving mg and ma into components I came out with:

g sin 22 = a cos 22

which gives me a = 3.96m/s^2.

Then using T = 2 pi (L/(g+a))^1/2

I found T = 1.42s, however the solution given in the book for this problem says T = 1.62s. Where have I gone wrong?

 

[OlderDan]

I prefer looking at it from the point of view of an observer who is not accelerating. The total force on the bob is ma (horizontal) The real forces acting on the bob that will give it this acceleration are the tension in the string and gravity. The sum of those forces must be in the direction of the acceleration. But you can, and I think you did, get the same result by treating ma as a mysterious force acting on the bob to give it a horizontal displacement. I think you went wrong when you added the two accelerations. Acceleration is a vector, and the two are not in the same direction.

 

[Elfrae]

Oh right, but does it matter if I use pseudo forces to solve this kind of problem, since it gets me the same answer for a? Is it just a matter of preference or should I be looking at the real forces instead?

I just tried it again adding the accelerations as vectors and got T = 1.62s, so hopefully I've got it right this time.

Thanks!

 

[OlderDan]

Oh right, but does it matter if I use pseudo forces to solve this kind of problem, since it gets me the same answer for a? Is it just a matter of preference or should I be looking at the real forces instead?

I just tried it again adding the accelerations as vectors and got T = 1.62s, so hopefully I've got it right this time.

Thanks!

No it does not matter as long as you handle the pseudo-forces correctly. Actually, for this problem it is probably better your way, since you are looking for an "effective" g acting in the direction of the string at equilibrium.