Simple Harmonic Motion and a Car

[stanli121]

Homework Statement

A car uniformly accelerates. Show an open door that slams shut will do so in simple harmonic motion.

Homework Equations

The Attempt at a Solution

This seems more conceptual than mathematical. I considered using T = Ia but the problem is the torque would act on the pivot (point where door connects to car) and be R=0 from the axis so T=0. Then I considered F=ma because the acceleration is known to be constant. But I ran into two problems there. First, the displacement I would need to relate to acceleration to show SHM is not clear whatsoever. Second, wouldn't the force about the pivot be centripetal, thus removing my x and d²x/dt²? My newest thought is perhaps choosing a point on the pavement and doing some sort of T = dL/dt type calculation then relating $$\theta$$ and $$\alpha$$ that way. Any comments on my ideas or new ideas?
Thanks.

 

[diazona]

You can calculate the torque around any axis. You're probably used to using pivot points because that's usually the convenient thing to do, but in this case you might want to consider other axes to try.

 

[kerimek]

You are correct in thinking that this problem is more conceptual than mathematical. To show simple harmonic motion in this scenario, we can consider the door as a pendulum. When the car accelerates, the door will swing open due to the inertia of the door. As the car continues to accelerate, the door will reach its maximum displacement and start to swing back towards the closed position. This back-and-forth motion of the door can be described as simple harmonic motion, with the car's acceleration acting as the restoring force.

To relate this to the equations of simple harmonic motion, we can consider the door's displacement from the closed position as the x-coordinate in our equation x = A*sin(ωt), where A is the amplitude and ω is the angular frequency. As the car continues to accelerate, the door's displacement will change sinusoidally, creating a smooth, back-and-forth motion.

In summary, the car's acceleration acts as the restoring force for the door, causing it to exhibit simple harmonic motion. This example highlights the wide applicability of simple harmonic motion in various physical systems.