Inclined accelerating pendulum question

[cloughenough]

Homework Statement

A pendulum with string length L hangs inside a boxcar that is rolling down an incline at an angle Beta. The pendulum swings to a maximum angle Theta. Ignore friction, assume Beta and Theta are small.

How far from the vertical will the pendulum swing to the left while the boxcar accelerates down the incline? (in terms of Theta and Beta).

The Attempt at a Solution

I assumed that since while accelerating, the new 'rest position' would be Beta left of vertical, that the pendulum would then swing Theta to the left and right of the new rest position. Thus, it would swing Beta+Theta to the left of the vertical.

The answer says : The frame of reference has been changed. In the new reference frame, the maximum angle from the new rest position is given by Beta+Theta (<--I don't get where Beta comes from!). This is how far it will swing to either side from the rest position. This means that it will swing to the left of vertical by an amount Theta + 2(Beta).

Thank you!

 

[tiny-tim]

Welcome to PF!

Hi cloughenough! Welcome to PF!

(have a beta: β and a theta: θ )

A pendulum with string length L hangs inside a boxcar that is rolling down an incline at an angle Beta.
…
In the new reference frame, the maximum angle from the new rest position is given by Beta+Theta (<--I don't get where Beta comes from!).

As you say, you have to prove that the new 'rest position' is β to the left of vertical.

To prove that, find the x and y components (in the stationary frame) of the acceleration of the boxcar.

Then combine that with the ordinary g to give the effective acceleration … what are its x and y components?

 

[nlightner]

I would like to clarify a few points in this question. First, it is important to specify the units for the angles Beta and Theta, as well as the acceleration of the boxcar. This will allow for a more accurate calculation and understanding of the problem. Secondly, it is important to define what is meant by "swinging to the left" - is this relative to the direction of motion of the boxcar or relative to the vertical? This will also affect the calculation.

Assuming that Beta and Theta are in radians and the acceleration of the boxcar is in m/s^2, the answer provided is correct. The angle Beta represents the incline of the boxcar, which is the new "rest position" for the pendulum. Therefore, the pendulum will swing Theta to the left and right of this new rest position, resulting in a total swing of Beta+Theta to the left of the vertical. This is because the pendulum will swing Theta to the left of the new rest position and then swing Theta to the right of the new rest position, resulting in a total swing of 2Theta. However, since the new rest position is already Beta to the left of the vertical, the total swing to the left of the vertical will be Beta+Theta.

In summary, the answer provided is correct and the explanation is accurate. However, it would be helpful to specify the units and clarify what is meant by "swinging to the left" in order to ensure a complete understanding of the problem.