- #1

brotherbobby

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- Homework Statement
- A plumb bob is hung from the ceiling of a train compartment. If the train moves with an acceleration ##a_0## along a straight horizontal track, the string supporting the bob makes an angle ##\tan^{-1}(a_0/g)## with the normal to the ceiling. Suppose the train moves on an inclined but straight track with uniform velocity. If the angle of incline is ##\tan^{-1} (a_0/g)##, the string again makes the same angle with the normal to the ceiling.

(a) Can a person sitting inside the compartment tell by looking at the plumb line whether the train is accelerated on a horizontal straight track or going uniformly up along an incline ?

(b) If yes, how ? If no, suggest a method to do so.

- Relevant Equations
- In an inertial frame, Newton's second law states that the net (external) force on a particle ##\Sigma F = ma##. This force is real - meaning it is due to physical interaction of the particle and its surroundings.

In a non-inertial frame that accelerates with an acceleration ##a_0## relative to an inertial frame, the net force acting on a particle of mass ##m## : ##\Sigma F = ma+ma_0##. Here the force is ##F_0 = ma_0## is not due to interactions - it is an apparent force observed in the non-inertial frame acting on the object.

(a) No, a person seated inside the train compartment

(b) I am assuming that both observers are not allowed to look "out" of the boundaries of the apartment. If they are, the former will notice surrounding objects accelerate with an acceleration ##a_0## for which no cause (real force) can be found. However, the latter observer will not notice any such accelerating bodies.

Assuming they are both confined to the boundaries of the compartment, I wonder if they can find out the motion of their trains by

For the second observer, the marble will roll to the "back" of the compartment with an acceleration ##a = g \sin \theta = \frac{g}{ \operatorname{cosec} \theta} = \frac{g}{\sqrt{1+\cot^2 \theta}} = \frac{g}{\sqrt{1+g^2/a_0^2}}\; (\text{since we are given}\;\tan \theta = a_0/g) = \frac{a_0}{\sqrt{a_0^2+g^2}}g\;\boxed{\neq a_0}##.

**will not be able to tell**whether the train is accelerating on a horizontal track or moving uniformly up an inclined track by observing the plumb line.(b) I am assuming that both observers are not allowed to look "out" of the boundaries of the apartment. If they are, the former will notice surrounding objects accelerate with an acceleration ##a_0## for which no cause (real force) can be found. However, the latter observer will not notice any such accelerating bodies.

Assuming they are both confined to the boundaries of the compartment, I wonder if they can find out the motion of their trains by

**letting a marble roll "down" the (frictionless) floor**. For the first observer, the marble will roll to the "back" of the train with an acceleration ##a_0##, equal to the acceleration of the train relative to ground.For the second observer, the marble will roll to the "back" of the compartment with an acceleration ##a = g \sin \theta = \frac{g}{ \operatorname{cosec} \theta} = \frac{g}{\sqrt{1+\cot^2 \theta}} = \frac{g}{\sqrt{1+g^2/a_0^2}}\; (\text{since we are given}\;\tan \theta = a_0/g) = \frac{a_0}{\sqrt{a_0^2+g^2}}g\;\boxed{\neq a_0}##.

**The different accelerations of the marble "down" the compartment floor will inform the observers as to status of their motion.**

Am I correct?Am I correct?