3.4.1.5 Newton’s laws of motion
Knowledge and application of the three laws of motion in appropriate situations.
$F = ma$ for situations where the mass is constant.
Newton’s 1st Law of motion
What causes motion? When things are moving what keeps them moving? It is easy to say that moving objects have kinetic energy, so does energy cause motion? How do objects gain that energy?
We can imagine an object like a ball being thrown, so the force from our arm starts the ball moving, so forces cause motion, but what happens to to the force from our arm on the ball once it has been thrown? Well once it has left our hands the force is zero, and the ball starts to slow down and follows a parabolic path towards the ground. Does this mean that if we remove the force then the motion will begin to stop? What if I threw the same ball in the vacuum of space? What path would it take then and for how far?
Galileo came up with the above through experiment when he was considering forces and motion. Imagine a ball being rolled down slopes in the absence of friction. How high would it reach on the second side? How far would it travel if the second side if the ramp was removed? You need to think carefully about these four diagrams as we will be discussing them in class.
When we consider the examples above we should come to the conclusion that forces are required to change motion, this is often hard to see in our world of friction and other resistive forces, but once a force has caused an object to start moving it will continue to move. This is the idea behind Newton’s 1st law of motion:
Every body continues in its state of rest or uniform motion in a straight line unless acted on by some external force.
When an object has a constant velocity the sum of the forces acting on it will be zero. It may sometimes be necessary to resolve the forces acting on the object into vertical and horizontal components to see this.
This law defines what a forces is and does. It causes acceleration It is also called the law of inertia, or the unwillingness of something to start moving, or to stop moving once it has started.
A body with a large mass requires a large force to change its speed or its direction, i.e. it has a large inertia. Therefore the mass of a body is a measure of its inertia.
This law is an extension of the idea of balanced forces that we met earlier when we learnt about statics, but is also applied to moving objects.
Newton’s 2nd law of motion
The 2nd law can be stated as:
The rate of change of momentum of a body is directly proportional to the external force acting on the body and takes place in the direction of the force.
This law defines the idea of a resultant force a force that can change the motion of an object. So when the sum of the forces is not zero, the object will accelerate.
Looking at this mathematically we can derive an equation from the above law.
Where $F$ is the applied force. and $\frac{\mathrm{d} }{\mathrm{d} t}\left ( mv \right )$ is the rate of change of momentum.
To turn this relationship into an equation we need to introduce a constant of proportionality, so:
The unit of force is defined so that one unit of force accelerates a mass of one kilogram by one metre per second squared so the constant k=1. This also makes it dimensionless so it requires no units.
We will also only consider situations where the mass remains constant, so we get:
We also know that $\frac{\mathrm{d}v }{\mathrm{d} t}$ is acceleration, so we now have an equation for the second law.
Remember that this describes the resultant force on the object, which may not always be the total force. It is important to look at all of the forces acting on a object and take them into account first.
Newton’s 3rd law of motion
You may be more familiar work this law in terms of, "every action has an equal and opposite reaction", however this statement is easily misunderstood, and therefore it is easy to get confused. I prefer to use the more formal expression below:
If a body A exerts a force on a body B, then B exerts an equal and oppositely directed force on A
This tells us that whenever there is one force, a second one also arises. We call these Newton pairs or action-reaction pairs. Using this statement it is clear that pairs of forces must alway act on different bodies. For example. A person in a lift. If the lift is stationary then the only forces acting on the person are weight W and the contact force from the floor N. In this situation it is true that W = N. But what if the lift is accelerating? Then there must be a resultant force acting on the person. As only W and N are acting N must now be larger than W so they cannot be an action-reaction pair.These forces are also acting on the same body, the person, so not body A and body B.
To help identify these pairs use the following checklist:
- Action reaction pairs must be the same size. (along the line of action)
- Action-reaction pairs always act on different bodies. otherwise nothing would ever happen as the forces would just cancel each other out!
- They must always be the same type of force, in the lift example W is a gravitational force and N is an electromagnetic force.
This idea can be difficult to get you head around, and will be discussed in detail in the classroom.