a car accelerating uniformly from 0 to 100kph in 10secs will have:
average acceleration = 100kph x 1000m / (3600secs x 10secs) = 2.78m/second2
momentum
linear momentum = mass x velocity
angular momentum = the moment of inertia (I) x angular velocity (ω)
moment of inertia (I) = mass x radius2 where radius is distance to the centre of motion
hence a ballerina will spin faster when limbs are close to the body as moment of inertia is reduced and thus angular velocity must increase as momentum remains constant
NB. momentum is conserved during collisions, explosions, and other events involving objects in motion, however, in the real world, some kinetic energy is lost to heat during a collision, especially if the collision is inelastic, and thus overall momentum will fall.
net system momentum is constant if the net external force (or for angular momentum, external torque) is zero
force
force = mass x acceleration = change in momentum / unit time
1 newton (N) = 1 kilogram X meter/second2
lifting a 1kg object to overcome gravity requires a force of 1kg x acceleration due to gravity = 9.8N
force to move a 2000kg car from 0 to 100kph in 10secs = 2000 x 2.78 = 5556N
kinetic energy
energy = 1/2 x mass x velocity2
1 Joule = 1 kg m2/sec2
gravitational potential energy
energy = mass x gravitational constant x height
mechanical energy (kinetic + gravitational) is CONSTANT unless there are non-conservative works done such as creation of heat or sound via friction, etc
work
work = force x distance = change in kinetic energy
1 Joule = 1Nm
work done to lift a 10kg object 1m off the ground = 10kg x 9.8 x 1m = 100J
work done to accelerate a 2000kg car from 0 to 100kph (excl. frictional and wind resistance work) = 0.5 x 2000 x (27.8m/s)2 = 27,800J
work done in an initially stationery object sliding down to the bottom of a ramp at an incline of theta and a friction coefficient of mu from height h and a length of the ramp L:
L = height / sin(theta)
work done = gravity work - frictional work = mgh - (mu x mgCos(theta) x L ) = change in KE of the object hence can calculate the end speed of the object at the bottom of the incline as 1/2 x mv2 = total work done value
power
power = work / time taken = energy used / time
1W = 1 Joule/sec
power to accelerate a 2000kg car from 0 to 100kph in 10sec = 27,800J / 10sec = 2780W
note that combustion engines are only about 40% efficient so the engine would actually need to use 2780/0.4 =7kW of power
petrol has a energy density of 44MJ/kg thus over 10sec at 7000W requires 70000J = 70kJ and thus ~44000/70 = 1.6mL of fuel IF there were no frictional forces to overcome such as wind resistance, etc.
rotating bodies
torque = moment of Inertia x angular acceleration
moment of Inertia for a solid disk = mass x radius2/2
eg. a falling single disk yo-yo suspended by the string contacting it's outer ring:
the torque of a falling yo-yo (assuming pure rolling without sliding) = tension in the string x radius of yo-yo (as the yo-yo rotates around the centre which is r distance from the string contact)
angular acceleration = linear acceleration / radius this gives tension = mass x linear acceleration / 2
now considering vertical forces of gravity and the opposing string tension:
(mass x gravity) - string tension = mass x linear acceleration, solving for string tension gives string tension = (mass x gravity) - (mass x linear acceleration)
substituting in tension = mass x linear acceleration / 2 gives mass x linear acceleration / 2 = (mass x gravity) - (mass x linear acceleration) gives linear acceleration = 2/3 x gravity
eg. cylinder of mass m and radius r rolling down a smooth incline of angle theta and unknown surface friction coefficient (mu) and assuming no slipping or sliding
to find the linear acceleration down the slope (a)
torque = moment of Inertia x angular acceleration and this is produced by the friction of the surface of the incline and the cylinder
vertical downwards gravitation force = mass x gravity
gravitation force in direction of the slope = mass x gravity x sin(theta)
gravitation force perpendicular to the slope = mass x gravity x cos(theta)
frictional force in direction opposite to the slope = surface friction coefficient (mu) x gravitation force perpendicular to the slope = surface friction coefficient (mu) x mass x gravity x cos(theta)
torque on the cylinder = frictional force x radius = surface friction coefficient (mu) x mass x gravity x cos(theta) x radius
moment of Inertia for a solid disk = mass x radius2/2
angular acceleration = linear acceleration / radius
substituting these in gives frictional force = 1/2 x mass x linear acceleration = surface friction coefficient (mu) x mass x gravity x cos(theta)
which gives surface friction coefficient (mu) x mass x gravity x cos(theta) = 1/2 x linear acceleration
hence mu = 1/2 x linear acceleration / (mass x gravity x cos(theta) )
force = mass x linear acceleration
force = downward gravitation force - frictional force = (mass x gravity x sin(theta) ) - (surface friction coefficient (mu) x mass x gravity x cos(theta) ) = mass x linear acceleration
hence (gravity x sin(theta) ) - (surface friction coefficient (mu) x gravity x cos(theta) ) = linear acceleration
eliminate mu by substituting in mu = 1/2 x linear acceleration / (mass x gravity x cos(theta) )
(mass x gravity x sin(theta) ) - (1/2 x linear acceleration / (mass x gravity x cos(theta) ) x mass x gravity x cos(theta) ) = mass x linear acceleration
which simplifies to: linear acceleration = 2/3 x gravity x sin(theta)
ie. the linear acceleration down the slope is INDEPENDENT on the surface friction coefficient (mu) AS LONG as the cylnder does not start slipping due to inadequate friction
capacity is measured in Watt-hours (Wh) ie. how many watts of power is available over 1 hour
watt-hours = amp-hours x nominal voltage
ie. a 100Ah 12.8V lithium battery will have capacity of 100 x 12.8 = 1280Wh and thus will last 12.8 hours if using 100W of power
Ohm's law
voltage difference between two points in a circuit (volts) = current (amps) x resistance (ohms)
heat
heat energy required to heat a substance
heat energy = mass x specific heat x change in temperature in deg C
eg. water has specific heat of 4,180J/kg°C
eg. cast iron and stainless steel has specific heat of ~500J/kg°C
Stefan–Boltzmann law for rate of radiant heat loss from a surface
radiant heat loss in watts = ϵσA(T14−T24)
where,
ϵ is the emissivity of the surface (dimensionless, between 0 and 1)
σ is the Stefan-Boltzmann constant (5.67×10−8 W m−2K−4)
A is the surface area (in square meters)
T1 is the absolute temperature of the radiating surface (in kelvin)
T2 is the absolute temperature of the surroundings (in kelvin)
Wien's displacement law for frequency of light emitted from a black body
peak frequency of light =2.82kBT/h
where,
kB is Boltzmann's constant (1.38×10−23J/K
T is the absolute temperature in kelvin
h is Planck's constant (6.63×10−34 J·s)
flow of fluids
laminar flow
there is minimal mixing between layers of the fluid and there is very little fluctuation of the speed of the flow
blood flow in arteries is generally laminar
turbulent flow
there is increasing mixing of the layers of flow with eddy formation resulting in varying flow rates - the greater the variation in flow, the greater the turbulence
energy cascade: turbulence converts kinetic energy into heat via the creation of ever smaller eddies and eventually dissipates due to viscosity
turbulence is complex and varies over both spatial and time dimensions
direct numerical simulation (DNS) can be used to model this but it is computationally complex to calculate, instead large eddy simulation (LES) is often used instead which uses a sub-grid scale model to address the small eddies
the Reynolds-averaged Navier-Stokes technique is the least computationally expensive model technique but this only models the effects of eddies using turbulent viscosity, not the actual eddies
given the computational complexity, wind tunnels are often used to assess the effect of wind on objects such as aerodynamic vehicles, tents, etc
turbulent flow requires more energy and results in a greater pressure drop
most flows in nature are turbulent
Reynold's number
a value for a given system which means turbulent flow is likely to occur if Reynold's number is high for a given system
Reynold's number (Re) = fluid density x velocity x characteristic length / dynamic viscosity = velocity x length / kinematic viscosity = inertial forces / viscous forces
for flow past a cylinder, characteristic length = cylinder diameter
for flow past a air foil (wing of an airplane), characteristic length = length
for flow through a pipe:
characteristic length = pipe diameter
turbulent flow is likely if Re > 2000-4000
in laminar flow, flow rate at the pipe wall is zero in “no-slip condition” and velocity increases to a maximum at the centre of the pipe and the velocity profile is parabolic
in turbulent flow, flow rate at the pipe wall is zero in “no-slip condition” (there is a laminar sublayer here but its thickness decreases as Re increases) however, away from the wall, flow rate is more even throughout the pipe due to the mixing of layers creating a more homeogenous time averaged flow velocity but with more random variations
if the pipe wall roughness thickness is less than the laminar sublayer thickness then the pipe is said to be hydraulically smooth as it will not effect the turbulent flow
Darcy-Weisbach equation to calculate pressure drop along a pipe
pressure drop = length of pipe x Darcy friction factor x (density/2) x (avg velocity)2/hydraulic diameter
for laminar flow, Darcy friction factor = 64/Re
for turbulent flow, there is a complicated Colebrook equation to determine the friction factor which requires an iterative calculation hence it is generally looked up on a Moody Diagram, but simplified pressure drop is proportional to velocity squared and pipe surface roughness