# The Physics of Disaster: An Exploration of Train Derailments [Excerpt]

Understanding the science behind trains can help identify the causes of accidents—and lead us to safer railways

Figure 7.1

the velocity vector, be it a change in speed or a change in direction, requires a force to create the change.

Consider a ball rolling along a straight line. One could constantly tap the ball with a stick, forcing it to move in a circular path. The tapping force, always pointing to the center, is changing the ball’s velocity vector’s direction. The ball is moving at a constant speed but changing direction; the ball is said to be accelerating toward the center of the circle.

Consider a 1-lb (0.45-kg) block rotating on the end of a 4-foot (1.2-m) string in a horizontal plane at a constant speed of 20 ft/sec (6 m/s). The direction of the velocity vector, always perpendicular to the string, is constantly changing and creating acceleration toward the center of rotation (Figure 7.1).

Acceleration for circular motion equals the velocity squared divided by the radius of the circle, or 100 ft/sec2 (30.5 m/s2). Per Newton’s Law, the string must exert a force on the block equal to m × a, or a force of 3.1 lbs (13.8 N) toward the center of rotation. (Recall that to correctly calculate f  m × a the weight must be converted into a mass by dividing by the acceleration of gravity—32.2 ft/sec2 [9.8 m/s2].) The force the string exerts on the block is called the centripetal, or center-seeking, force. The block exerts an inertial load on the string, keeping it tight. The so-called centrifugal force is not a force; it’s the block’s inertial resistance to the centripetal acceleration. The 1-lb (0.45-kg) block resists the acceleration imposed by the string just as the person in the elevator

Figure 7.2.

resists the upward acceleration. The term centrifugal force is incorrect. We will use the term centrifugal inertial loading. But, of course, the so-called centrifugal force feels like a force when holding on to the string attached to the rotating block.

A locomotive moving around a curve is similar to a rotating block on the end of a string. Both experience acceleration toward the center of rotation. The inertial loading keeps the string tight and creates a lateral force on the locomotive at the wheels. Lateral forces between the wheels and the rail must react against the centrifugal inertial loading to keep the train on the tracks.

If the centrifugal inertial loading is excessive, the locomotive begins to tip. The ﬂange of the wheel catches on the rail and the locomotive starts to rotate, as shown in Figure 7.2. In fact, that’s why the ﬂanges are on the inside of the wheels. If the ﬂanges were on the outside, the slightest bit of wheel lift would slide the locomotive off the tracks.

In the 1947 Pennsylvania Railroad overturning accident, the locomotive weighed 320,000 lbs (145,150 kg). The centrifugal inertial loading of the locomotive moving at 88 ft/sec (60 mph [97km/h]) on a curve with a radius of 675 feet (206 m) is:

Horrible number photo

The centrifugal inertial loading is trying to tip the locomotive clock-wise about the pivot point (the bottom of the right wheel). This rotation is resisted by the weight of the locomotive (also acting through its center of gravity), which tries to rotate the locomotive counterclockwise.

The locomotive’s weight and inertial load both exert a torque. A torque is a twisting force applied to the end of a lever arm that tries to tighten a nut. A 10-lb (44.5-N) force on the end of a 9-inch (23-cm)-long wrench exerts a torque of 10 × 9 = 90 inch lbs of torque (10 Nm).

The Pennsylvania locomotive had a center of gravity 80 inches (2 m) above the rail. The centrifugal inertial loading tries to rotate the locomotive with a clockwise torque equal to 114,000 lbs × 80 inches—more than 9 million inch lbs of torque (6.3 × 106 Nm).

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1. 1. phalaris 03:07 AM 8/4/13

Good background article for the technically interested (i.e. probably the majority of SciAm readers!). Many thanks.

Shame the site is being polluted by these spammers again.
I don't see it on other blogs: it seems odd that SciAm can't prevent it.

2. 2. plcsys 05:04 PM 8/8/13

Thank you for the really interesting and educational article. It goes a long way to understanding the modes of failure for rail cars. I wonder how many derailments are from rail misalignment rather than overspeed on a curve. I have often wondered if a slow moving long freight train can derail towards the inside of the curve because of the force transmitted through coupling tension.

3. 3. Satya Narayan Tiwary 07:16 PM 8/11/13

Speed may cause derailment on curve because centrifugal force depends on square of speed.
S. N. Tiwary
Director

4. 4. Satya Narayan Tiwary 07:17 PM 8/11/13

Speed may cause derailment on curve because centrifugal force depends on square of speed.
S. N. Tiwary
Director

5. 5. pbalant 10:39 AM 8/19/13

A good article on train dynamics. The trains mentioned in this article derailed due to overspeed. It mentioned Postive Train Control, also known as Automatic Train Control. These systems drive the trains by computer control that can perform these mundane tasks much more reliably and safely than human drivers. The systems also protect from collisions and other hazards. In fact, there are commuter railways that have been successfully run without drivers since 1986.

6. 6. Gord Davison in reply to plcsys 07:25 PM 12/1/13

I would expect that many derailments have been blamed on excessive speed which were probably caused by poor rail and locomotive maintenance. The rail system is a profit oriented organization and things like maintenance are a cost that is always trying to be reduced.

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