Physics Learning Lessons (Classical Mechanics)-III

Newton’s Laws of Motion

Portrait of Isaac Newton
by Kneller in 1689

Newton’s First Law

A body in motion remains in motion unless it is acted on by an external force. If the body is at rest it remains at rest.

Newton’s first law may seem counter intuitive to our everyday experience. If we move an object by pushing it, it will move and then come to a halt. To keep it moving at the same speed a force must be applied. However, most objects are slowed to a halt by friction or air-resistance. Newton’s first law gives rise to the idea of inertia. It takes a large force to start to move an object with a large mass and it also takes a large force to stop it moving. A moving object has a reluctance to stop moving and a stationary object has a reluctance to move.

An external force is a force that is applied to the object. There may be other forces inside the object, for example, there are forces produced by the motion of the atoms that make up the object but on average, they cancel each other out and so do not effect the motion of the object.

Newton’s first law applies in inertial reference frames, i.e. when the frame of reference is stationary or moving with constant velocity. It does not apply when the frame of reference is accelerating. Strictly speaking, the Earth is accelerating because it is rotating as well as orbiting the Sun, however the effect of this is small enough to consider the Earth as an inertial reference frame.

Newton’s Second Law

The force, F on an object is equal to the rate of change of momentum, p, with time.

In mathematical terms.

F = dpdt(1)

Or since p = mv, then

dpdt = d dt (mv) = m dvdt + vdmdt (using the product rule)

For constant mass, dm/dt = 0 is zero therefore,

F= mdvdt

We recognise dv/dt as the acceleration, a and so an alternative form of Newton’s 2nd law is

F = ma(2)

and since, a = dvdt = d²rdt²

F = md²rdt²(3)

For this relationship to be true, we must measure the force F in Newtons [N], the mass m in kilograms and the acceleration in ms^-2 otherwise there will be a constant of proportionality will have a value other than 1.

The force in Newton’s 2nd law refers to the resultant force. If there are many forces acting on an object, it is the vector sum of those forces is the force that produces any acceleration.

∑F_ext = ma(4)

Newton’s Third Law of Motion

Newton’s third law applies to pairs of bodies. If a body A exerts a force on a body B, then body B exerts an equal and opposite force on body A.

Newton’s third law reminds us that forces occur in equal and opposite pairs.

There are many examples of Newton’s third law all around us. Every object that is on the floor or on top of another object is exerting a weight force toward the center of the Earth. If the object is stationary, then whatever the object is on top of is exerting an equal force in the opposite direction.

A rocket is also a good example of Newton’s third law, it works by accelerating a mass in the form of hot gases, which produces a force. This produces an equal and opposite fore that pushes the rocket in the opposite direction.

The Importance of Newton’s Laws

Newton’s laws are extremely important not just in mechanics but in the whole of physics. When trying to understand a physical process, we often understand it by looking at the forces acting and working out the equations of motion. This is true of the motion of the planets to the flow of electrons in an electric or magnetic field.

Work

We often use words in physics that have a different a meaning from their everyday useage. Work is a good example of this. The scientific definition is

Work, is the product of the resolved force F, acting in the direction of motion, and the displacement through which the object travels.

If the force, F does not cause the object to move, then no work is done. A man holding a heavy weight in the air is not doing any work in the scientific scense of the word, although it might be hard work to keep it in the air, which is converting chemical energy of the muscles into thermal energy by muscular action. Mathematically speaking,

Where F is the applied force, s is the displacement and θ is the angle between the force acting on the object and the direction of motion. Work is a scalar because of the dot product between two vectors.

If the force acts in the same direction as the motion, then θ is 0 degrees then the work becomes, W=|F||s|, since cos (0)=1. Conversely, if the force is at right-angles to the displacement then θ= 90° and sin (90)=0;. Therefore force has no component in the direction of the displacement. i.e. the forces are independent. Therefore, the work done by the force is zero.

All this assumes the force acting on the object is constant with time. If the force varies with the distance then the work done could be found by plotting a graph of the force in the direction of motion against the displacement. For a small enough displacement the work done can be approximated by δW=F(s)δs.

The total work is then simply the sum of all the strips. As δ goes to zero then the sum becomes the integral. Therefore a new definition of work is;

When the environment does work on a system, W > 0; that is, the total energy of the system increases. Work is done on the system.

When a system does work on its environment, W < 0; that is, the total energy of the system decreases. Work is done by the system.

Work is done by the system when the resolved force has a component in the same direction as the displacement. If the resolved force has a component in the opposite direction to the displacement, the work done is negative since cos θ <1 for θ > 90 °. For example when two surfaces slide over each other the work done by the frictional forces is always negative, since the frictional force opposes the relative motion.

For a variable force, the work done can be found by plotting the component of force in the direction of the displacement, as a function of the displacement. The work done is the area under the curve.

The unit of work is the [N.m]. This derived unit is named after John Prescot Joule who did much work on the conversion of energy from one form of to another. A Joule is defined as:

One Joule is the work done by a force of 1 Newton when its point of application moves through a distance of 1 metre in the direction of the force. 1 [J]= 1 [N.m]