# Pendulum in an accelerating frame

• Elfrae
In summary, the conversation discusses the use of a yo-yo as a simple pendulum to measure acceleration of a jet plane during takeoff. The speaker presents their method of solving the problem by considering total force on the bob and resolving it into components, resulting in an acceleration of 3.96m/s^2. However, the solution given in the book is 1.62s. The speaker then questions the use of pseudo forces and whether it affects the accuracy of their solution. They later try solving the problem by adding the accelerations as vectors and arrive at the same answer as the book's solution. The other person confirms that using pseudo forces is a valid approach as long as it is handled correctly, and suggests it may be
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?

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 accleration 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.

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!

Elfrae said:
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.

## 1. How does a pendulum behave in an accelerating frame?

In an accelerating frame, the pendulum will appear to swing at an angle instead of in a straight line. This is because the frame of reference is constantly changing and the pendulum is influenced by this acceleration.

## 2. What causes a pendulum to behave differently in an accelerating frame?

The acceleration of the frame of reference is the key factor in causing the pendulum to behave differently. This acceleration can be caused by various forces, such as gravity or centrifugal force.

## 3. Does the length of the pendulum affect its behavior in an accelerating frame?

Yes, the length of the pendulum does affect its behavior in an accelerating frame. A longer pendulum will have a longer period of oscillation and will appear to swing at a slower rate compared to a shorter pendulum.

## 4. How does the acceleration of the frame affect the period of a pendulum?

The acceleration of the frame does not affect the period of a pendulum. The period of a pendulum is determined by its length and the force of gravity, and is independent of the acceleration of the frame of reference.

## 5. Can a pendulum ever reach equilibrium in an accelerating frame?

No, a pendulum will never reach equilibrium in an accelerating frame. This is because the acceleration of the frame is constantly changing the direction and magnitude of the force acting on the pendulum, preventing it from coming to rest.

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