IB recommended length: 10 hours
Section A.1 Kinematics looked at how we can describe the motion of an object. This section examines the forces that influence that motion and allow us to make predictions about what will happen. We will look at how to represent forces both visually and algebraically. We will dive into the mathematics of Newton's laws of motion, still a useful way to model motion even if superseded by Einstein in detail, and we will look at the relationship between forces and moment in making predictions about the motion of objects.
A force is a a push or a pull acting on an object. Forces are a vector quantity, they have a magnitude, measured in newtons, N, and always have a direction. The standard symbol for a force is the letter F, but this may be changed or modified using subscripts if we need to specify a particular type of force.
As vectors, forces can be described using words (e.g. "a 5N force acting to the left") or using diagrams. Alternatively we can draw accurate diagrams (free-body diagrams) that illustrate ll of the forces acting on a body or a simple system of bodies. In these diagrams forces are represented by arrows. The placement and direction of an arrow precisely indicate the point of application and the direction of application of a force, and the length of the arrow represents, according a chosen scale, the magnitude of the force. All forces in free-body should be labelled using commonly accepted names or symbols.
If many forces act on the same body, free-body diagrams can be analysed to determine the resultant (overall or net) force acting on the body. It is this overall force that will allows to make predictions about the acceleration and future behaviour of the body.
All forces are identical in terms of their consequences, a 20N for acting North will have the same influence on a body no matter the origin of the force, but it is useful to identify types of forces. In classical Newtonian mechanics, all forces represent interactions between a pair of objects. However, it is useful to classify and categorise forces. We have contact forces, any force that is a result of two objects in contact with each other, and field forces, forces that act between object separated in space. The idea of and mathematical development of fields as in gravitational or magnetic fields postdates Newton, being developed by Faraday and James Clerk Maxwell.
The normal force, FN, is a contact force that acts perpendicular when a force is exerted on a solid body. This force is electromagnetic in origin. For example, when leaning against a wall you are pushing the molecules of the wall into each other. These molecules are separated by bonds with a natural length and compressing those bonds will induce a force pushing back. The combined effect of all those forces is the normal force.
The surface frictional force, Ff, acts parallel to the plane of contact between two surfaces in contact. If the two surfaces are not moving relative to each other, the the static frictional force will cancel the inducing force up to a maximum value, the product of the normal force and the coefficient of static friction, μs. So, Ff ≤ μs FN. If the maximum static frictional force is exceeded then the surfaces will start moving relative to each other and the dynamic friction force, equal to the product of the normal force and the coefficient of dynamic friction, μd, will be in operation. Ff = μd FN.
The elastic restoring force, FH, that applies to all stretched materials and is described by Hooke's Law. FH = - kx, where x is the extension, the amount the material is stretched by in metres, and k is the spring constant, how much force is required to stretch by 1m, so units of N/m.
The vicious drag force, Fd, acting on a small sphere (as an approximation of real objects) as it moves through a fluid. The equation is Fd = 6πηrv. Where η is the fluid viscosity, r is the radius of the sphere and v is the velocity of the sphere through the fluid.
The buoyancy force, Fb , that acts on a body immersed in fluid with a pressure gradient due to the displacement of the fluid. Fb = ρVg, where ρ is the density of the fluid in kg/m³, V is the volume of the body in m³ and g is the local force of gravity.
The following field forces must be known and used:
The gravitational force, Fg, that gives an object weight and is calculated by Fg = mg, where m is the mass of the object in kilograms and g is the local gravitational field strength - on Earth 9.81 N/kg (numerically the same as the acceleration due to gravity at the surface of the Earth). We will look at gravitational fields in more detail in D.1 Gravitational fields.
The electric force, Fe. This force acts on any object with an electrical charge if the local electrical field is not zero. For more details on the electric force see D.2 Electric and magnetic fields.
The magnetic force, Fm. This force acts on any object with a magnetic dipole if the local magnetic field is not zero. For more details on the electric force see D.2 Electric and magnetic fields.
Momentum is a basic property of all matter. All moving objects with a non-zero mass have momentum. Momentum is a vector quantity, the symbol for momentum is a lower-case p and the basic formula for momentum is:
p = mv
where m is mass in kilograms and v is the velocity of the object in m/s. The unit for momentum is thus kilograms.metres per second, kg.m/s.
Unless an external force acts on the system (i.e. you don't have an isolated system) then the total momentum of the system is a conserved quantity.
An external force that does act on an object will change the momentum of the object. We say that the change in momentum of the object, symbol J, is called the impulse.
J = F∆t = ∆p = ∆mv
where F is the average applied force during the impulse in newtons and ∆t is the time taken for the impulse in seconds. The units of impulse are either newton.seconds, Ns, or the unit for momentum, kilogram.metres per second, kg.m/s.
Rearranging the formula for impulse gives us an equation relating force, impulse and the rate of change of momentum.
F = ∆p/∆t = ∆mv/∆t
Newton's three laws of motion were first stated in his book, Principia Mathematica, published in 1687 (although developed much earlier). They form the basis of classical mechanics. The laws can be summarised as:
Newton's first law: A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force. This law is embodied in the concept of inertia. An object moving at a constant speed in a straight line is said to be in translational equilibrium.
Newton's second law: When a body is acted upon by a force, the time rate of change of its momentum equals the force. This is a re-statement of the above equation. Alternatively, for constant mass situations, if F = ∆mv/∆t, then F = m.∆v/∆t, so F = ma, or force = mass × acceleration. Newton's second law is really an expansion of Newton's first law, with the mathematics that lets us make predictions.
Newton's third law: on the otherhand, telss us something about the nature of forces. Namely that forces care an interaction between two objects, thus forces come in pairs (a force-pair) and that whenever a force acts on one object then a second force, of equal magnitude but opposite direction, acts on the other object.
Momentum is conserved in all situations. Energy is always conserved in all situations, at least in classical mechanics. We will look at energy in more detail in the next section, but for now, as you should know from previous work, there are two main categories of energy: potential energy, Ep, and kinetic energy, Ek. All energy is a scalar quantity and there are various equations for potential energy depending on the situation, but for kinetic energy the basic equations is Ek = 1/2.mv².
When two objects collide then momentum and total energy will be conserved in an isolated system, but the kinetic energy may not be conserved.
Elastic collisions conserve kinetic energy. Collisions between rigid bodies such as billiard balls, or atoms in the kinetic theory of gases, are good approximations to elastic collisions.
Inelastic collisions do not conserve kinetic energy. Instead some of the energy goes into other forms, such as in deforming the objects or emitting sounds, although eventually all of the 'lost' energy is degraded into heat energy.
Explosions are extreme inelastic collisions. If all the components of an explosion form an isolated system, then they have no kinetic energy before the explosion, just potential energy in the form of elastic, chemical or some other potential energy. The total momentum is also zero as there is no motion. After the explosion, some or all of the potential energy is converted to kinetic energy. The total momentum remains zero because momentum is a vector quantity and the explosion parts are travelling in various directions opposite to each other. Explosions in one dimension (such as the release of a spring connecting two trollies) need to be considered by all students, but two-dimension explosions only need concern HL students.
All the above concepts have applied to motion in a straight line, linear motion, but we must also consider objects moving in circles. Motion along an ellipse or an oval is generally approximated to circular motion at this level.
Centripetal acceleration is experienced by all objects moving along a circular path. Since all such objects are changing direction, they are also changing velocity (since velocity is motion with a speed and a direction). Any change of velocity is an acceleration. Acceleration is also a vector quantity. The direction of acceleration is towards the centre of the circular path. If the speed, v, (such as would be shown on the speed-dial of a car) is constant then the equations of centripetal acceleration are:
a = v²/r
where v is the linear velocity in m/s, r is the radius of curvature of the curved path in metres (m).
Angular velocity, ω, is a property of an object in circular motion and describes the rate of change of an angle connecting the object in circular motion with the centre of the circle. The SI units of angular velocity are radians per second, although degrees per second is also used. The period, T, of an object in circular motion is the time taken for one complete revolution in seconds. The acceleration of an object in uniform circular motion can be written using these terms as:
a = v²/r = ω²r = 4π²r/T²
Newton's second law for uniform circular motion, assuming constant mass, is just F = ma with one of the above expressions for acceleration substituted in. Most commonly:
F = mv²/r
Since all objects in circular motion are accelerating there must always be a centripetal force that acts in the direction of the centre of the circle. For uniform circular motion this must be the only overall force acting on the object and will be perpendicular to the linear velocity. The actual nature of the centripetal forces depends on the situation: e.g. for a planet orbiting a star it will be gravity, for an ion in a mass spectrometer it will be the electrical force. If the magnitude of the velocity is not constant then the body will change direction even if it does not result in uniform circular motion.
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