Analyizing the dynamics of a pendulum hanging in an accelerating car

[gatechphysics]

Homework Statement

A small weight of mass 'm' hangs from a string in an automobile. Initially, the car is at rest with the weight hanging vertically. Then, the car SUDDENLY accelerates to a rate 'A'. Find the maximum angle $$\varphi_{max}$$ through which the weight swings by analyzing the motion of the pendulum WITHIN AN INERTIAL REFERENCE FRAME.

Homework Equations

In my book they derive the solution to this problem using an analysis within a NON-INERTIAL reference frame. The answer is:

$$\varphi_{max}$$ = 2*arctan(A/g) (1)

If you instead try to find the static angle the pendulum makes with the vertical in a car that is already accelerating at a constant rate A you will find that the static angle is half of the maximum angle given by equation (1).

The problem we are being asked though is to try and derive equation (1) using a non-inertial frame and I cannot figure out how to carry out the calculation.

The Attempt at a Solution

I am assuming that the best way to solve this problem is to use a classic Newton's 2nd Law analysis using an inertial reference frame that is stationary with respect to the car. Or is it better to solve this problem using an energy approach? Note that when I say an energy approach I do NOT mean forming the Lagrangian and solving that way. In my class we have only been doing classical analyses using Newton's formulation of mechanics so Lagrangian and Hamiltonian analyses are not acceptable for my class. By energy approach I simply mean that the weight increases in potential energy as it approaches the max angle, thus the corresponding increase in the kinetic energy of the bob must equal the increase in grav. potential so maybe there is a way to connect the kinetic energy of the bob with the acceleration of the car?

By the way this is not a homework question. Instead my professor challenged us to attempt to solve this problem in an inertial frame. I hope that I am not posting this question in the wrong forum. If I am I am sorry but I am new here so just let me know and I will move it.

 

[nickjer]

In the part 2 you say "derive equation (1) using a non-inertial frame" which contradicts what you said in part 1. So I am confused now. Basically what specific reference frame are you deriving this in... Outside of the car, inside of the car, or the bob's frame of reference.

 

[gatechphysics]

In the part 2 you say "derive equation (1) using a non-inertial frame" which contradicts what you said in part 1. So I am confused now. Basically what specific reference frame are you deriving this in... Outside of the car, inside of the car, or the bob's frame of reference.

Sorry. It should be "inertial frame". In my textbook the solution is given for a non-inertial frame. The challenge was posed to us to try to solve this problem within an inertial frame. Thanks for letting me know about that typo. I will edit it right away.

 

[gatechphysics]

1. Homework Statement

A small weight of mass 'm' hangs from a string in an automobile. Initially, the car is at rest with the weight hanging vertically. Then, the car SUDDENLY accelerates to a rate 'A'. Find the maximum angle $$\varphi_{max}$$ through which the weight swings by analyzing the motion of the pendulum WITHIN AN INERTIAL REFERENCE FRAME.

Homework Equations

In my book they derive the solution to this problem using an analysis within a NON-INERTIAL reference frame. The answer is:

$$\varphi_{max}$$ = 2*arctan(A/g) (1)

If you instead try to find the static angle the pendulum makes with the vertical in a car that is already accelerating at a constant rate A you will find that the static angle is half of the maximum angle given by equation (1).

The problem we are being asked though is to try and derive equation (1) using a inertial frame and I cannot figure out how to carry out the calculation.

The Attempt at a Solution

I am assuming that the best way to solve this problem is to use a classic Newton's 2nd Law analysis using an inertial reference frame that is stationary with respect to the car. Or is it better to solve this problem using an energy approach? Note that when I say an energy approach I do NOT mean forming the Lagrangian and solving that way. In my class we have only been doing classical analyses using Newton's formulation of mechanics so Lagrangian and Hamiltonian analyses are not acceptable for my class. By energy approach I simply mean that the weight increases in potential energy as it approaches the max angle, thus the corresponding increase in the kinetic energy of the bob must equal the increase in grav. potential so maybe there is a way to connect the kinetic energy of the bob with the acceleration of the car?

By the way this is not a homework question. Instead my professor challenged us to attempt to solve this problem in an inertial frame. I hope that I am not posting this question in the wrong forum. If I am I am sorry but I am new here so just let me know and I will move it.

It wouldn't allow me to edit my original post so please note the corrected sentence in bold in the above quote.

 

[nickjer]

I thought the inertial reference frame was outside the car and not inside the car since that frame was accelerating. I only ask because you also say "I am assuming that the best way to solve this problem is to use a classic Newton's 2nd Law analysis using an inertial reference frame that is stationary with respect to the car."

Also, I recommend drawing a free body diagram to help you solving this.

 

[gatechphysics]

I thought the inertial reference frame was outside the car and not inside the car since that frame was accelerating. I only ask because you also say "I am assuming that the best way to solve this problem is to use a classic Newton's 2nd Law analysis using an inertial reference frame that is stationary with respect to the car."

Also, I recommend drawing a free body diagram to help you solving this.

I don't understand your question. If the frame was inside the car it would cease to be an inertial frame. The question is to solve the problem in an inertial frame. Not necessarily any particular inertial frame. Given this, I was simply assuming in the quote above that an inertial frame that was stationary with respect to the car (e.g. a reference frame attached to the surface of the Earth which sees the pendulum and car accelerating past at a rate A) was the best inertial frame to work within. This assumption might be incorrect.