DYNAMIC AEROPLANES

Figure 3.1 shows a modern aeroplane. To decrease cost and the effect on the environment, such an aircraft must reduce air resistance and weight, yet be able to use air resistance and other forces to stop when landing. If you have ever flown in an aeroplane you will know how the back of the seat pushes you forwards when the aeroplane accelerates down the runway. The pilot must control many forces on the aeroplane in take-off, flying and landing.
In Chapters 1 and 2 we saw how motion can be described in terms of displacement, velocity, acceleration and so on. Now we are going to look at how we can explain how an object moves in terms of the forces that change its motion.
Apart from air resistance, see how many other forces you can discover that act on an aeroplane.
Compare your list with someone else. What causes all these forces?

Figure 3.2a shows how we represent the force that the motors on a train provide to cause it to accelerate.
The resultant force is represented by a green arrow. The direction of the arrow shows the direction of the resultant force. The magnitude (size) of the resultant force of $20000 N$ is also shown.

To calculate the acceleration a of the train produced by the resultant force F, we must also know the train’s mass m (Table 3.1). These quantities are related by:

$a = \frac{F}{m}$ or $F = ma$

In this example, we have $F = 20000 N$ and $m = 10000 kg$, and so:

$a = \frac{F}{m} = \frac{{20000}}{{10000}} = 2\,m\,{s^{ - 2}}$

In Figure 3.2b, the train is decelerating as it comes into a station. Its acceleration is $ - 3.0\,\,m\,{s^{ - 2}}$. What
force must be provided by the braking system of the train?

$F = ma = 10000 \times  - 3 =  - 30000N$

The minus sign shows that the force must act towards the right in the diagram, in the opposite direction
to the motion of the train.

Newton’s second law of motion

The equation we used, $F = ma$, is a simplified version of Newton’s second law of motion: For a body of constant mass, its acceleration is directly proportional to the resultant force applied to it.
An alternative form of Newton’s second law is given in Chapter 6, when you have studied momentum.
Since Newton’s second law holds for objects that have a constant mass, this equation can be applied to a train whose mass remains constant during its journey.
The equation $a = \frac{F}{m}$ relates acceleration, resultant force and mass. In particular, it shows that the bigger the force, the greater the acceleration it produces. You will probably feel that this is an unsurprising result. For a given object, the acceleration is directly proportional to the resultant force:

$a \propto F$

The equation also shows that the acceleration produced by a force depends on the mass of the object. The mass of an object is a measure of its inertia, or its ability to resist any change in its motion. The greater the mass, the smaller the acceleration that results. If you push your hardest against a small car (which has a small mass), you will have a greater effect than if you push against a more massive car (Figure 3.3).
So, for a constant force, the acceleration is inversely proportional to the mass:

$a \propto \frac{1}{m}$

The train driver knows that when the train is full during the rush hour, it has a smaller acceleration. This is because its mass is greater when it is full of people. Similarly, it is more difficult to stop the train once it is moving. The brakes must be applied earlier to avoid the train overshooting the platform at the station.