Simple mechanics problem.

Question

Question

A train is travelling West to East along a straight horizontal track. A child suspends a pendulum from the roof of one of the carriages and notices that the pendulum is inclined at an angle of 4◦ to the vertical. Calculate the acceleration of the train.

Where I am at so far:

Let the mass of the bob be m.

I don't know where to start, I don't even know if my diagram is correct.

T is the tension in the string.

Answer 1

Start with balance of forces projected: $$ \begin{align} ma = T \cos(90^\circ - 4^\circ)\\ mg = T \sin(90^\circ - 4^\circ) \end{align} $$

Answer 2

You know two things:

1. The vertical component of $T$ must equal the weight of the bob, since the bob is not accelerating vertically. So:

$T \cos(4^o)=mg$

2. The horizontal component of $T$ must be the force required to accelerate the bob horizontally at the same acceleration $a$ as the train (since the bob is stationary with respect to the train). So:

$T \sin(4^o)=ma$

You can eliminate $T$ and $m$ from these two equations and find an expression for $a$ in terms of $g$.

Answer 3

Suppose the pendulum hangs at an angle of $\theta$. Then, we can equate horizontal and vertical forces to give $ma = T \sin \theta$ and $mg = T \cos \theta$. This yields $m = \frac{T}{a} \sin \theta$, so $\left ( \frac{T}{a} \sin \theta \right )g = T \cos \theta$, so $\boxed{a = g \tan \theta}$.