Mechanics Tutorial 10 - Newton’s Laws of Motion

 Contents

Newton’s First Law states:

Every object continues in its state of rest or uniform motion in a straight line, unless it is compelled to change that state by an external force acting on it.

A car will maintain a constant speed if the drive force and the drag are balanced.   The total force is zero.

 The graph below shows the acceleration of a car up to its maximum speed. (a)  Why is the graph a straight line at low speeds? (b) Use Newton I to explain why the car reaches a maximum speed which it cannot exceed.  Assume it’s on a test track, so it can exceed the 115 km h-1 national speed limit.

Newton II

Newton's Second Law states:

Rate of change of momentum is proportional to the total force acting on a body, and occurs in the direction of the force.

An object of mass, m, is acted on with a constant force, F, so that its velocity increases from an initial value, u, to a final value, v, in time, t.

Momentum is the product of mass and velocity.  It is a vector and has units of kilogram metres per second (kg m s-1).

Change in momentum = momentum at end - momentum at start

Therefore:

Therefore

Now we know that acceleration:

So we can write:

Therefore:

(The term k is a constant with no units. k = 1)

Force (N) = Mass (kg) × acceleration (m s-2)

F = ma

Make sure you use the right units:

• Force in N,

• Mass in kg,

• Acceleration in m s-2.

Acceleration is always caused by a total force, or the resultant force, the vector sum of all the forces.  The acceleration is always, without exception, in the same direction as the total force.

Strictly speaking, we should write the equation as:

SF = ma

The strange symbol S is Sigma, a Greek capital letter 'S'.  It is physics code for "sum of".  Consider two coplanar forces F1 and F2 acting at right angles.  To get SF, we need to do a vector sum on the two forces:

 Question 2 A 70 kg athlete accelerates to his maximum speed of 9.5 m s-1 in a time of 2.5 s.  What is the average force he applies to the track?

If you have a vehicle providing a force to accelerate another vehicle, you must add the mass of the towing vehicle to that being towed.

 Question 3 A locomotive of mass 100 tonnes is hauling a train of wagons of mass 1200 tonnes with a pulling force (tractive effort) of 180 kN.  What is the acceleration of the train? (1 tonne = 1000 kg) Question 4 A horse of mass 800 kg is pulling a barge of mass 5000 kg as shown in the diagram below: The barge is initially at rest. (a) Calculate the acceleration. (b) The horse reaches a maximum speed of 1.5 m s-1.  Calculate the time taken to reach this speed. Question 5 (Challenge) This is part of an examination question A trolley of mass 1.0 kg is set up on a friction compensated runway, and attached to a 1 N slotted mass, which is released.   The trolley moves down the runway, and its motion is recorded with a ticker tape timer. Further slotted masses are added, and the motion measured. Describe how this experiment can be used to verify Newton’s Second Law of Motion. Explain why this is a bad question.
Newton III

Newton’s Third law states that:

If body A exerts a force on body B, body B must exert an equal and opposite force on body A.

In other words, forces always act in pairs.  This is true whether the forces are in equilibrium, moving, stationary or accelerating.

Source not known

This boy is sitting on a simple hovercraft.  The motor drives a fan which forces air downwards onto the floor. The force of the air going down produces a reaction force that lifts the machine off the ground.  A simplified diagram of the machine is shown below:

This picture shows a small hovercraft that can skim on water or dry land.  It is powered by a small (10 kW) petrol motor.

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Hover mowers are very simple small hovercraft.  Some transport hovercraft are enormous machines of mass several hundred tonnes.  They move easily across flat ground.  The problem with hovercraft is that their performance on hills is hopeless.  Can you think why?

Picture by Andrew Berridge, Wikimedia Commons

 Newton I and III are often confused.

Inertia (Extension only)

Let's go back to Newton I:

Every object continues in its state of rest or uniform motion in a straight line, unless it is compelled to change that state by an external force acting on it.

Now consider a space probe in deep space moving at 50 km s-1.  The sum of all the gravity acting on the probe is zero.  There is no force from the probe's motors.  There are no other forces to act on the probe.  However the probe has a constant velocity of 50 km s-1.

It may sound counter-intuitive that the probe keeps on going at a constant speed.  It should slow down and stop.  That happens here on Earth.  if we look at the second part of Newton I, the key part is, "unless it is compelled to change that state by an external force acting on it".  The external forces acting on an object here on the ground include friction and drag.  These are not present in space.

This leads us to the concept of inertia, which is the resistance of an object to change in its state of motion.

Consider this situation:

A ball bearing is released on the left hand side to run down the side of a perfectly smooth bowl.  If there is no friction or drag, the ball bearing should reach the equivalent spot on the other side, and the height from the bottom will be exactly the same.  This shows that the ball bearing has inertia, i.e. resistance to change in motion.

We know that all the energy has been converted from gravitational potential to kinetic, and back to gravitational potential.  Therefore:

Ep = Ek

mgDh = 1/2 mv2

Normally we would cancel out the mass on each side.  But the mass is involved in the transfer of the potential to kinetic energy and back.  Remember that the moving bearing was not subject to any external forces, so the quantity involved was the inertia.  So we can say the inertia has something to do with the mass.  And this is the case:

Mass is the quantity that solely depends on the inertia.

The more inertia a mass has, the greater the mass.  We then can say that mass is the amount by which an object resists change of motion.  Strictly speaking we should call this mass inertial mass.  Gravitational mass is mass involved with gravity fields.  Experiments show that both are identical.

You may find this video-clip helpful:

To sum up:

• Mass is a measurable quantity, while inertia is a concept that that explains how hard it is to change the current state of mass.

• In classical physics, mass is a property of an object, while inertia is a property of motion of the object, as well as the mass.

• Inertia is a concept that describes mass.

All of this may seem to be pedantic and inertia is one of those topics that physicists get really wound up with.  On the physics Quora website that I referred to in preparing this note, there is a range of views from a variety of contributors.  Many physicists go back to the idea that mass is "the amount of material in an object", which is perfectly good for most purposes.  However in rotational mechanics, inertia in its form of moment of inertia is an important concept to explain the behaviour of spinning objects.