# The physics behind tilting trains

So I'm trying to get an intuition behind the physics behind banked curves (and tilting trains). I know how to do the calculations, but I'm still struggling with the "why".

If I'm sitting in a train, and the train is engineered in such a way that it tilts $\theta$ degrees during a turn, how would you draw the free body diagram of a person sitting in the train in such an event? Normally, the normal force $N$ would be $\cos(\theta)mg$ (assuming there's a tilt but no velocity). However, when you introduce velocity and curves into the equation the normal force $N$ suddenly becomes larger than $mg$, i.e. $$N = mg/\cos(\theta) .$$

What gives? Obviously it has something to do with the centripetal acceleration, but I feel like total moron for not understanding the intuition behind it.

• Vectorially add the centripetal force and the force of gravity. – CuriousOne May 31 '15 at 15:05
• @CuriousOne Does the person in the train feel the centripetal force, or simply the friction? – user1904218 May 31 '15 at 15:29
• What friction are you talking about? The person in the train feels both forces, that of gravity and the force that results in a circular motion (without that force the train and its passengers would keep going straight). – CuriousOne May 31 '15 at 15:48
• Side note: Tilting trains do less than banked curves! Tilting trains are just for passenger comfort, banked curves allow higher velocities due to higher normal forces on the rails. – Sebastian Riese May 31 '15 at 16:00

This is a difficulty I see regularly: people often confuse real forces with the direction of the acceleration they experience.

In your situation, the real forces are gravity and the interaction with the seat/floor. The interaction with the seat/floor is often split into two components, tangent to the contact point (friction) and perpendicular to the contact point (normal). People often try to assume they know the normal force, but they can't without doing a proper free-body diagram. A good rule is do not assume that the normal force magnitude is equal to the weight. (Often, in overly-simplified drill problems it is, but in the real world it usually is not.) Let it have its own symbol ($F_N$ or $\mathcal{N}$) and see what the analysis tells you about how it relates to other quantities.

Draw a diagram (another failure on the part of many) showing the directions of these two/three forces. Then, superimpose a coordinate system that is convenient. The choice of the coordinate system never affects what happens, but does affect how one describes what is happening. If you are dealing with something moving circularly in a horizontal plane, choose horizontal and vertical, even if the train is tilted. The normal force from the seat/floor will be tilted, as will the friction, but you must resolve those into horizontal and vertical components so that you can apply Newton's 2nd Law to each coordinate direction easily.

Choose the positive horizontal direction to be toward the center of the circular path. Those force components will sum and be equal to the mass times the centripetal acceleration, in whatever form you want (linear speed, angular speed, period, etc): $$\Sigma F_{\mathrm{horizontal}} = ma_{\mathrm{centripetal}}$$ The vertical force components sum to equal the mass times the vertical acceleration (zero for horizontal planar motion): $$\Sigma F_{\mathrm{vertical}} = ma_{\mathrm{vertical}}=0$$

When you solve this system, you may be able to write an equation $$\mathcal{N}=\text{some combination of other quantities}.$$

Remember that this is a specific relationship between the magnitude of the normal force and other things affecting the object. It is not a general statement that explains the normal force, nor is it a fundamental equation.

Stand Up. Do you feel your weight being directly attracted to the floor (normal) ?
In a tilted train, in a curve, you feel the same: directly attracted to the floor (and not to the wall).

Of course that your seat (and the body of the train) have managed to change in a way they are 'bellow' you.
The vector sum of the vectors in presence....

1. normal to the Earth surface, gravity acc.
2. parallel to the Earth surface, centripetal acc

is pointing to the center of your seat.