Pendulum in an accelerating car

1. Nov 13, 2013

andyrk

1. The problem statement, all variables and given/known data

When a car is accelerating with a pendulum hanging inside it, then why does the pendulum get aligned with the vertical at an angle? Can you explain it without using the concept of pseudo force? I tried it but I was not able to get it.

3. The attempt at a solution
My Attempt:

According to what has been taught in the class, for determining which way the pendulum gets aligned, we assume any one direction and then resolve the tension force in the string in 2 directions. One along the motion of the car and the other upward, towards the roof of the car.
Using the horizontal direction force we can judge which way the pendulum should actually align using Newton's Second Law of Motion F is proportional to a (acceleration). But following this the pendulum should align in the forward direction, i.e the right direction if the car is accelerating towards right. But its the opposite, i.e the pendulum aligns leftwards and not rightwards. Can anyone explain why this happens?

2. Nov 13, 2013

Tanya Sharma

Hi andyrk...

Consider a small ball of mass 'm' hanging by a light string in the car .

Suppose the car moves horizontally towards right with acceleration 'a' . What should be the horizontal acceleration of the bob after it has come at rest with respect to the car ?

3. Nov 13, 2013

Mentz114

You are correct that the equilibrium position of the pendulum is found by resolving the forces. I think the reason why the bob lags behind is because its inertia is trying to keep it in the same place, and what we see is the (equal and opposite) reaction to the forward force.

4. Nov 13, 2013

andyrk

But that is again involving a Non Inertial frame of reference. Can't this be explained via inertial frame of reference, i.e the ground?

5. Nov 13, 2013

Tanya Sharma

No...I am explaining this using inertial frame only .

The ball accelerates horizontally with acceleration 'a' . The only way it can do so is if there is a component of force in the horizontal direction .This is possible only when the ball makes an angle θ with the vertical such that Tsinθ provides the necessary horizontal force .The ball has to be towards left side of the vertical line .

This is not possible if the ball is towards right or if it stays making zero angle with the vertical .

6. Nov 16, 2013

andyrk

Hmm..Yes you are right. Thanks.

7. Nov 16, 2013

lucasem_

Consider two accelerations being applied via forcers: Gravity, and the acceleration of the car. The sum of the forces (which act in different directions) will give you and effective acceleration, say g'. This is NOT directly downwards, which explains why the pendulum is in equilibrium at an angle to the vertical

8. Nov 16, 2013

haruspex

The acceleration of the car is not a force - it is a result of forces. Your argument amounts to use of pseudo-forces, which was ruled out in the OP.

9. Nov 16, 2013

lucasem_

The acceleration of the car is a force, inertially, because the pendulum is mounted at a hinge which is accelerating at $a_\text{car}$

10. Nov 16, 2013

WannabeNewton

Not it isn't. First of all that isn't even dimensionally consistent. Secondly, the acceleration of the car gives rise to an inertial force in the rest frame of the car but not only is this is a use of non-inertial frames (which the OP said was not a desired method) but also it involves the use of a force that vanishes in the lab frame that the OP wants to work in.