Inclined accelerating pendulum question

In summary, a pendulum with string length L hangs inside a boxcar that is rolling down an incline at an angle Beta. When the boxcar accelerates down the incline, the pendulum will swing to the left of the vertical by an amount of Theta + 2(Beta) in the new reference frame. This is because the new 'rest position' is Beta to the left of the vertical, and the maximum angle from this new rest position is given by Beta+Theta. To prove this, one can find the x and y components of the acceleration of the boxcar and combine them with the ordinary g to give the effective acceleration in the stationary frame.
  • #1
cloughenough
1
0

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!
 
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  • #2
Welcome to PF!

Hi cloughenough! Welcome to PF! :smile:

(have a beta: β and a theta: θ :wink:)
cloughenough said:
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? :wink:
 
  • #3


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.
 

What is an inclined accelerating pendulum?

An inclined accelerating pendulum is a physical system that consists of a bob attached to a string or rod, which is then attached to a fixed point on an incline plane. When the incline plane is accelerated, the bob experiences a change in its position and velocity.

What factors affect the motion of an inclined accelerating pendulum?

The motion of an inclined accelerating pendulum is affected by its mass, the angle of the incline plane, the acceleration of the incline plane, and the length of the string or rod.

How is the acceleration of an inclined accelerating pendulum calculated?

The acceleration of an inclined accelerating pendulum can be calculated using the formula a = g(sinθ - μcosθ), where a is the acceleration, g is the acceleration due to gravity, θ is the angle of the incline plane, and μ is the coefficient of friction between the bob and the incline plane.

What happens to the motion of an inclined accelerating pendulum when the angle of the incline plane is increased?

As the angle of the incline plane is increased, the acceleration of the pendulum also increases. This causes the bob to move faster and cover a greater distance in a shorter amount of time.

Can an inclined accelerating pendulum reach a point of equilibrium?

No, an inclined accelerating pendulum cannot reach a point of equilibrium as long as the incline plane is being accelerated. The motion of the pendulum will continue to oscillate due to the changing acceleration of the incline plane.

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