Newton's Law

Newton's Second Law

"Change of motion is proportional to the force applied, and take place along the straight line the force acts."

Newton's second law for the gravity force - weight - can be expressed as

F = m g (1)

where

F = force (weight)

m = mass

g = acceleration of gravity

The force caused by gravity - g - is called weight. Note! Mass - m - is a property.

The acceleration of gravity can be observed by measuring the change of velocity of a free falling object:

g = dv / dt (2)

where

dv = change in velocity

dt = change in time

A dropped object accelerate to a speed of 9.81 m/s or 32.174 ft/s in one second.

Acceleration of Gravity in SI Units

g = 9.81 m/s²

Acceleration of Gravity in Imperial Units

g = 32.174 ft/s²

Velocity and Distance Traveled of a Free Falling Object

The velocity of a free falling object can be expressed as:

v = g t (3)

where

v = velocity

The distance traveled by a free falling object can be expressed as:

s = 1/2 g t² (4)

where

s = distance traveled by the object

The velocity and distance traveled by a free falling object:

Note! The velocity is achieved without any aero-dynamical resistance (vacuum). The air resistance will be significant for higher velocities or for object with large surface area to mass ratio - feathers or similar.

Newton's First Law

"Every body continues in a state of rest or in a uniform motion in a straight line, until it is compelled by a force to change its state of rest or motion."

Newton's Third Law

"To every action there is always an equal reaction - if a force acts to change the state of motion of a body, the body offers a resistance equal and directly opposite to the force."

Concrete Properties

Properties of normal strength Portland cement concrete

· Density : 2240 - 2400 kg/m³ (140 - 150 lb/ft³)

· Compressive strength : 20 - 40 MPa (3000 - 6000 psi)

· Flexural strength : 3 - 5 MPa (400 - 700 psi)

· Tensile strength : 2 - 5 MPa (300 - 700 psi)

· Modulus of elasticity : 14000 - 41000 MPa (2 - 6 x 10⁶ psi)

· Permeability : 1 x 10^-10 cm/sec

· Coefficient of thermal expansion : 10^{-5 o}C^-1 (5.5 x 10^{-6 o}F^-1)

· Drying shrinkage : 4 - 8 x 10^-4

· Drying shrinkage of reinforced concrete : 2 - 3 x 10^-4

· Poisson's ratio : 0.20 - 0.21

· Shear strain : 6000 - 17000 MPa (1 - 3 x 10⁶ psi)

· Specific heat capacity : 0.75 kJ/kg K (0.18 Btu/lb_m ^oF (kcal/kg ^oC))

Frictional force can be expressed as

F_f = μ N (1)

where

F_f = frictional force (N, lb)

μ = static (μ_s) or kinet

Basic Information you may need

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Poisson's ratio is the ratio of the relative contraction strain, or transverse strain normal to the applied load, to the relative extension strain, or axial strain in the direction of the applied load.

Can be expressed as υ = - εt / εl (m/m) (1)

Where: υ = Poisson's ratio

εt = transverse strain

εl = longitudinal or axial strain

Strain can be expressed as ε = dl/L (2)

Where: dl = change in length

L = initial length

For most common materials the Poisson's ratio is in the range 0 - 0.5.

Material	Poisson's Ratio
Upper limit	0.5
Aluminum	0.334
Aluminum, 6061-T6	0.35
Aluminum, 2024-T4	0.32
Glass	0.24
Stainless Steel 18-8	0.305
Steel, cast	0.265
Steel, Cold-rolled	0.287
Steel, high carbon	0.295
Steel, mild	0.303
Concrete	0.2

Stress can be expressed as stress = F / A (2)

Stress = (N/m²) (lb/in², psi)

F = force (N) (lb)

A = area of object (m²) (in²)

Young's Modulus (Tensile Modulus)

Young's modulus or Tensile modulus can be expressed as

E = stress / strain = (F / A) / (dL / L) (3)

Where

E = Young's modulus (N/m²) (lb/in², psi)

Hooke's Law

One of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law which can be expressed as

F_s = -k dL (4)

Where

F_s = force in the spring (N)

k = spring constant (N/m)

dL = elongation of the spring (m)

Yield strength

Yield strength, or the yield point, is defined in engineering as the amount of strain that a material can undergo before moving from elastic deformation into plastic deformation.

Ultimate Tensile Strength

The Ultimate Tensile Strength - UTS - of a material is the limit stress at which the material actually breaks, with sudden release of the stored elastic energy.

1 N/m² = 1x10^-6 N/mm² = 1 Pa = 1.4504x10^-4 psi

1 psi (lb/in²) = 144 psf (lb_f/ft²) = 6,894.8 Pa (N/m²) = 6.895x10^-3 N/mm²

Modulus of Rigidity (or Shear Modulus) is the coefficient of elasticity for a shearing force. It is defined as "the ratio of shear stress to the displacement per unit sample length".

Modulus of Rigidity can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material.

Definition of Modulus of Rigidity:

The ratio of shear stress to the displacement per unit sample length

1 GPa = 0.145x10⁶ psi (lb_f/in²)

1 x 10⁶ psi = 6.9 GPa

The Bulk Modulus Elasticity can be expressed as

E = - dp / (dV / V) (1)

Where E = bulk modulus elasticity

dp = differential change in pressure on the object

dV = differential change in volume of the object

V = initial volume of the object

The Bulk Modulus Elasticity can be alternatively expressed as E = dp / (dρ / ρ) (2)

Where: dρ = differential change in density of the object

ρ = initial density of the object

An increase in the pressure will decrease the volume (1). A decrease in the volume will increase the density (2).

The SI unit of the bulk modulus elasticity is N/m² (Pa)

The imperial (BG) unit is lb_f/in² (psi)

1 lb_f/in² (psi) = 6.894 10³ N/m² (Pa)

A large Bulk Modulus indicates a relative incompressible fluid.

Bulk Modulus for some common fluids can be found in the table below:

Material	Shear Modulus
	(106 psi)	(GPa)
Aluminum Alloys	3.9	27
Aluminum, 6061-T6	3.8	24
Aluminum, 2024-T4	4	28
Carbon Steel	11.2	77
Concrete	3	21
Glass, 96% silica	2.8	19
Plywood	0.09	0.62
Structural Steel	11.5	79.3
Stainless Steel	11.2	77.2
Steel, Cast	11.3	78
Steel, Cold-rolled	10.9	75

Mass and Weight - the difference! - What is weight and what is mass? An explanation of the difference between weight and mass.

Specific Weight can be expressed as

γ = ρ g (2)

where

γ = specific weight (N/m³)

ρ = density (kg/m³)

g = acceleration of gravity (m/s²)

The SI-units of specific weight are N/m³. The imperial units are lb/ft³. The local acceleration g is under normal conditions 9.807 m/s² in SI-units and 32.174 ft/s² in imperial units.

Example - Specific Weight Water

Specific weight for water at 39 ^oF (4 ^oC) is 62.4 lb/ft³ (9.81 kN/m³) in imperial units. Specific weight in SI units can be calculated like

γ = (1000 kg/m³) (9.81 m/s²)

= 9810 N/m³

= 9.81 kN/m³