Mechanics

**Link to page on teaching ideas for mechanics**

Some key terms defined:

 * Mass** - the amount of stuff in an object, or, a measure of an object's inertia. Measured in kilograms (kg).


 * Inertia** - is the reluctance of an object change its velocity. Inertia depends on mass (only). Think: you can push a toy car with a small force, whereas a large car may require all your strength.


 * Force** - a push or a pull. The force that acts on an object is measured in newtons (N). Force is the work done on an object each second. Remembering that F=ma, Newtons are equivalent to kg m s -2.


 * Momentum** - A car (1 500 kg) moving at 5 ms -1 (18 km/h) would hurt you if it hit you. A hockey ball (0.02 kg) moving at 27 ms -1 (100km/h) would also hurt! Why does a small thing (low mass) moving quickly (high velocity) hurt like a big thing (high mass) moving more slowly (low velocity). This is to do with momentum. Momentum (p) = mass (m) x velocity (v). The greater an object's momentum, the more force is required to stop it (see Newton's second law of motion below). Measured in kgms -1


 * Weight** - a special name for a particular force: the force of gravity acting on an object. A typical value for the force acting on 1kg on the Earth's surface is 9.81N (often approximated to 10N). So a //mass// of 10kg will have a //weight// of about 100N at the Earth's surface. In every day speaking however, we talk of an objects weight in kilograms. This is incorrect, but as the acceleration due to gravity on the surface of the Earth is approximately constant (9.81ms -2 ) the weight in N is a linear function of mass, so changes in mass cause a directly proportional change in weight (on or near the Earth's surface). Measured in N.


 * Density** - measures mass per unit volume (kg/m 3 ). Note for calculations: to convert from cm 3 to m 3, multiply by 10 6 (100x100x100). SI unit is r (rho).

//Note: // When considering fields, g (the gravitational field strength) is defined as the force per unit mass (N kg-1 ). An object in free fall is being acted on by a force g, and has a resulting acceleration of g in m s-2. So g (N/kg) is equivalent to m/s2.


 * Forces on solids **

Stress = Force (N) per unit area ( m 2 ), where the force is perpendicular to the area (at 90 degrees) Strain = increase in lenght (D L) per unit lenght (L)

Young modulus = Stress / Strain (N/m 2 ) - note that this is also the unit of pressure (pascal, Pa)

Note that stress-strain graphs are usually drawn with the stress (independent variable) on the y-axis (against the usual rule of plotting the independent variable on the x-axis). This is done so that the gradient of the line is the Young Modulus (at least over the range where Hooke's Law (extension is proportional to the stretching force - i.e. where the line is straight) is obeyed).

Newton's Laws of Motion
The early Greeks, including Aristotle, thought that the natural state of objects was that they were stationary, and that all objects tended to that position. So you had to apply a force to get something moving, but once you stopped applying the force it would try to return to being stationary. This is in line with every day experience: if you push a supermarket trolley away from you it will slow down and come to a rest (hopefully before it hits another shopper!)

Galileo, in the late 16th/early 17th centuries proposed instead that a object would retain its motion if there was no force (e.g. friction) acting on it. He determined this based on kinematic analysis (analysis of the motion geometrically), without consideration of the forces acting (dynamic analysis).

Newton, in the mid to late 17th century developed these ideas further, including developing a new mathematics which forms the basis of modern day calculus, and proposed three law of universal motion which could be used to predict, from known initial conditions, the future motion of bodies with a level of accuracy what was not improved upon until Einstein some 200 years later (and which makes material improves to Newton's laws only at velocities close to the speed of light).


 * Newton's First Law (aka the Law of Inertia)**


 * If there is no //resultant// force acting on an object it will remain at rest or, if it is moving, it will continue moving with a constant velocity.

Note: "resultant" is a very important part of the law, as all every day objects will have forces acting on them, gravity, for instance. The desk in from of me is not however moving due to the force of gravity as there is no resultant force acting on the desk. A resultant force is a force that is not cancelled out by an equal but opposing force acting on the //same object//.

This can be demonstrated with an air-track so that frictional forces are all but eliminated. As the link below shows, letting friction back into the picture makes a big difference to the motion of the air-track trolley.

[|Air-track demo (University of Sydney)]

This is also know as the Law of Inertia, as inertia is an objects reluctance to change its state of motion, which is a restatement of the first law.


 * Newton's Second Law**


 * The force acting on an object equals the rate of change of momentum with time, and the change in momentum takes place in the direction of that force.

In other words, the rate of change of momentum of an object is directly proportional to the resultant force acting on it. So the more momentum that an object has, the more force is required to stop it.

Note that if an objects mass is unchanging, then as F = (mv-mu)/dt, we can write:

F = mdv/dt

as acceleration = dv/dt

F=ma, which his how this law is often written.


 * Newton's Third Law**


 * For every action there is an equal but opposite reaction.

This law is typically remembered as the above, but this can lead to a misunderstanding of the law. For example, if any action (force) has an equal and opposite reaction (force in the opposite direction) they must cancel out, so there is no resultant force and nothing changes? This is not what the law says, and the equal but opposite force of the law applied to a //different body//. The law can therefore be better remembered as:


 * If object A exerts a force on object B, then B exerts an equal but opposite force on object A.

This can be thought about in a number of ways:

1) Shopping trolley in space

2) Shopping trolley in a car park

3) Two spring force-meters (one attached to a desk, and one in your hand, attached together).

Practically, it is important to distinguish between the free-body forces and the total system forces.