Physics of a roller coaster

 

The exciting roller coaster!

Fig. 1  The exciting roller coaster!

The journey of the roller coaster.

Fig. 2  (a) The roller coaster is brought to the highest point. (b) The potential energy of the coaster decreases, but the speed and the kinetic energy increase. (c) After passing through the "valley", the height of the coaster increases, but the speed and the kinetic energy decrease. (d) Due to frictional lost, the mechanical energy of the coaster has decreased, so the second "hill" has to be lower than the first one.

The roller coaster experiences a centripetal force.

Fig. 3  According to Newton's 1st law, if there is no external force, the roller coaster would move uniformly in a direction tangential to the rail.

The equilibrium of forces.

Fig. 4  The equilibrium of forces when the roller coaster is at the highest position.

When you are sitting on a roller coaster, traveling at a speed of about 100 kilometers per hour, will you think that why it doesn't need any engine? Why don't we fall down when we are looping? (Why can you still sitting in front of the computer and browsing this web page?)

The first question is concerned about the conservation of energy. Energy can be in different forms, such as kinetic energy, potential energy, sound energy and heat etc. When you are sitting on the roller coaster, because of the lift motor, it will be lifted to a very high position. Relative to the ground, the roller coaster has a very large potential energy (Fig. 2). The higher position it is, the larger potential energy it has. After passing the highest position, it goes down the hill, both the speed and the kinetic energy increase. By conservation of energy, the potential energy will decrease and transform into kinetic energy.

When the roller coaster has just passed the first "valley", it has a lot of kinetic energy, so it can climb up the second "hill". By conservation of energy, it can climb up to a height equal to that it went down before. However, due to the friction in the machines, the total mechanical energy of the roller coaster will decrease. As a result, the first "hill" of many roller coasters are the highest, but the followings will have decreasing heights.

The second question is related to centripetal force, whose principle is more complicated. According to Newton's first law of motion, a body will continue in its state of uniform motion along a straight line when there is no net force acting on it. That means if the roller coaster is under no external forces, it would move in a straight line tangential to the track (Fig. 3). However, there should be forces acting on it because its direction of motion changes. Assuming that the mass of the roller coaster is , it is moving at a velocity , and the track is circular with a radius , the centripetal force required would be (Fig. 4). But where does this force come from?

For simplicity, let's consider the case when the roller coaster is at the highest point. If it has a very high speed at this point, the centripetal force for turning will be larger. The weight of coaster supplies part of the centripetal force, but if the required centripetal force is larger than the weight (), then the rest of the centripetal force must be supplied by the reaction force acted by the rails on the coaster. Adding up these two parts, we have . The larger velocity the coaster has, the larger is the interaction force between the rails and the coaster, consequently the coaster will be more tightly attached to the rails and will not fall down. On the contrary, if the velocity is smaller, the rails will give smaller or even no reaction force, and the roller coaster will risk falling down. In other words, because of the high speed, the roller coaster will not fall down.

Let's try a simple experiment. Prepare a rope and a paper cup. Tight the edge of the cup with the rope, and then half fill the cup with water. When you are swinging the cup, making it to move in a vertical circle (like a cowboy), you will discover that no water spills out! It is because the bottom of the cup supplies a centripetal force to the water.