Non-inertial reference frame, a pendulum in an accelerating car 
Take a pendulum that is suspended in a car; it hangs from a rope in a car. When the car starts to accelerate with acceleration $a_0$ to the right, from an inertial frame,

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*Why does the pendulum move to the left?


*Why does the pendulum appear to fall back, i.e. the pendulum tilts at an angle $\theta$ with respect to the vertical?
Why is this example so confusing? Can anyone explain what is happening from the perspective of both inertial and non-inertial frames?
 A: First imagine what would happen if the system were in outer space without gravity. The bob would be floating around, and when the car accelerates to the right, the top of the rope would be pulled to the right, drawing more rope after it until the rope were pulled taut horizontally with the bob trailing along at its left end.
That's what you're seeing here, except there's also gravity accelerating the bob downward, so the angle is where the acceleration A and the acceleration of gravity are balanced.
It's a tricky thing to imagine gravity as an acceleration UPWARD, but it's totally equivalent to that, exactly like an elevator going up. Let's say the outer space car (no gravity) is on a platform that's accelerating up like an elevator. The car accelerates to the right by amount A, and it also accelerates up by amount g. So the car is actually accelerating on a diagonal made by the hypotenuse of A and g. Up and to the right. So the bob gets "pulled" down and to the left.
If you replace the upward outer-space elevator with just ordinary gravity on Earth, then the bob does the same thing: pulled down and to the left.
A: I interpret your question as 'why does this tilt back when car is accelerating?'.
With respect to the ground frame (inertial), the car is accelerating. And the pendulum is inside the car. Won't that move with the car? Definitely it will. To do so, there must be a net horizontal force on the bob. But when car is at rest, there are only vertical forces acting on the bob, tension of the string and the weight. So to gain a horizontal acceleration (when the car starts accelerating), it slightly tilts backwards, because then it can produce a horizontal force with the component of the tension along the string which helps the bob to accelerate.
Let's go to non-inertial frame. You can always find fictitious forces in non-inetial frames. The fictitious force heads to the opposite direction which the system is accelerating. In this incident, the fictitious force acts on the bob backwards. Thus it will pull the bob.
Hope this helps.
A: REALITY
Gravity is pulling down on the ball, and the top of the string is moving, which puts a force along the direction of the string in tension, up and to the right. A string can only provide force along its direction, in tension. So down, plus up and right, can only result in the ball accelerating to the right from a still frame.
INERTIAL FRAMES INTUITUON
How does this force work inside the inertia frame? I always remember that if a room is falling, it is accelerating downward and has no overall force inside the room. Like zero gravity. So acceleration down makes apparent force up, which cancels the gravity force down. Accelerating down at the acceleration of gravity provides an upward-seeming force equal to the force of gravity.
INERTIAL FRAMES STRATEGY
The key is get the one apparent gravity (by adding the vectors) and forget youre moving and forget which part is actual gravity and which part is inertial, and then work on the problem.
INERTIAL FRAME THIS PROBLEM
Imagine we are inside the car. To us, it is like gravity is pointing to the lower left, and stronger than normal, instead of straight down. Lower left because it is the vector sum of the downward gravity and the leftward centripetal force. If the acceleration of the car is very high, there is (apparently inside the car) an almost completely leftward overall force.
A string can only provide instantaneous force along its own direction, in tension. So if the acceleration is high enough that the direction of our apparent “gravity” is more than theta from vertical, then the net force in our frame will be perpendicular to the string making it swing left, to us. (The tension in the string cancels out any aspect of force in that direction, leaving only force perpendicular to the string. Thats how tension works.) So add g vector and -a vector and see if it is more or less than theta from vertical.
