A scalar is a quantity with a size, for example mass or length A vector has a size (magnitude) and a direction.
Velocity is the rate and direction of change in position of an object. Velocity is the rate and direction of change in position of an object. For example, at the beginning of the Winter Break, our car had an average speed of 61.39 miles per hour, and a direction, South. The combination of these two properties, speed and direction, forms the vector quantity Velocity
Vectors can be broken down into components For example in two dimensions, we can define two mutually perpendicular axes in convenient directions, and then calculate the magnitude in each direction Vectors can be added The brown vector plus the blue vector equals the green vector
Acceleration is the change in Velocity during some small time interval. Notice that either speed or direction, or both, may change. Acceleration is the change in Velocity during some small time interval. Notice that either speed or direction, or both, may change. For example, falling objects are accelerated by gravitational attraction, g. In English units, the speed of falling objects increases by about g = 32.2 feet/second every second, written g = 32.2 ft/sec2
Most scientists and engineers try to avoid English units, preferring instead SI units. For example, in SI units, the speed of falling objects increases by about 9.81 meters/second every second, written Most scientists and engineers try to avoid English units, preferring instead SI units. For example, in SI units, the speed of falling objects increases by about 9.81 meters/second every second, written g = 9.81 m/sec2 Unfortunately, in Hydrology our clients are mostly civilians, who expect answers in English units. We must learn to use both.
Hydrologists will take 1/5th of Geol. jobs. Hydrologists will take 1/5th of Geol. jobs. Petroleum Geologists make more money, 127K vs. 80K, but have much less job security during economic downturns. Hydrologists have much greater responsibility. When a petroleum geologist makes a mistake, the bottom line suffers. When a hydrologist makes a mistake, people suffer.
Hydrologists provide numbers to engineers and civil authorities. Clients ask, for example: Hydrologists provide numbers to engineers and civil authorities. Clients ask, for example: “When will the crest of the flood arrive, and how high will it be?” “When will the contaminant plume arrive at our municipal water supply?
In your work as a hydrologist, you will be scrounging for data from many sources. It won’t always be in the units you want. We convert from one unit to another by using conversion factors. In your work as a hydrologist, you will be scrounging for data from many sources. It won’t always be in the units you want. We convert from one unit to another by using conversion factors. Conversion Factors involve multiplication by one, nothing changes 1 foot = 12 inches so 1 foot = 1 12 “
Water is flowing at a velocity of 30 meters per second from a spillway outlet. What is this speed in feet per second? Water is flowing at a velocity of 30 meters per second from a spillway outlet. What is this speed in feet per second? Steps: (1) write down the value you have, then (2) select a conversion factor and write it as a fraction so the unit you want to get rid of is on the opposite side, and cancel. Then calculate. (1) (2) 30 meters x 3.281 feet = 98.61 feet second meter second
The product of velocity and area is a flow rate The product of velocity and area is a flow rate V [meters/sec] x A [meters2] = Flow Rate [m3/sec] Notice that flow rates have units of Volume/ second It is very important that you learn to recognize which units are correct for each measurement or property.
Water is flowing at a velocity of 30 meters per second from a spillway outlet that has a diameter of 10 meters. What is the flow rate? Water is flowing at a velocity of 30 meters per second from a spillway outlet that has a diameter of 10 meters. What is the flow rate?
Water is flowing at a rate of 3000 meters cubed per second from a spillway outlet. What is this flow rate in feet3 per hour? Water is flowing at a rate of 3000 meters cubed per second from a spillway outlet. What is this flow rate in feet3 per hour? Let’s do this in two steps 3000 m3 x 60 sec x 60 min = 10800000 m3/hour sec min hour 10800000 m3 x (3.281 feet)3 = 381454240. ft3/hr hour ( 1 meter) 3
Momentum (p) is the product of velocity and mass, p = mv Momentum (p) is the product of velocity and mass, p = mv In a collision between two particles, for example, if there is no friction the total momentum is conserved. Ex: two particles collide and m1 = m2, one with initial speed v1 , the other at rest v2 = 0,
Force is the change in momentum with respect to time. Force is the change in momentum with respect to time. A normal speeds, Force is the product of Mass (kilograms) and Acceleration (meters/sec2), So Force must have SI units of kg . m sec2 1 kg . m is called a Newton (N) sec2
If all forces and Torques are balanced, an object doesn’t move, and is said to be static If all forces and Torques are balanced, an object doesn’t move, and is said to be static
Discussion Torques, See-saw Reference frames Discussion Dynamics
Pressure is Force per unit Area Pressure is Force per unit Area So Pressure must have units of kg . m sec2 m2 1 kg . m is called a Pascal (Pa) sec2 m2
Density is the mass contained in a unit volume Thus density must have SI units kg/m3 The symbol for density is pronounced “rho” Very important is not a p, it is an r It is NOT the same as pressure
Suppose you need the density of water in kg/m3. You may recall that 1 cubic centimeter (cm3) of water has a mass of 1 gram. Suppose you need the density of water in kg/m3. You may recall that 1 cubic centimeter (cm3) of water has a mass of 1 gram. 1 gram water x (100 cm)3 x 1 kilogram = 1000 kg / m3 (1 centimeter)3 (1 meter)3 1000 grams water = 1000 kg / m3
Mass Flow Rate is the product of the Density and the Flow Rate Mass Flow Rate is the product of the Density and the Flow Rate i.e. Mass Flow Rate = AVelocity Thus the units are kg m2 m = kg/sec m3 sec
Energy is the ability to do work, and work and energy have the same units Work is the product of Force times distance, W = Fd Distance has SI units of meters 1 kg . m2 is called a N.m or Joule (J) sec2 Energy in an isolated system is conserved KE + PE + Pv + Heat = constant
Energy has Energy has units kg . m2 sec2 So pressure energy must have the same units, and Pressure alone is kg . m sec2 m2 So if we multiply Pressure by a unit volume m3 we get units of energy
Kinetic Energy (KE) is the energy of motion KE = 1/2 mass . Velocity 2 = 1/2 mV2 SI units for KE are 1/2 . kg . m . m sec2
Potential energy (PE) is the energy possible if an object is released within an acceleration field, for example above a solid surface in a gravitational field. Potential energy (PE) is the energy possible if an object is released within an acceleration field, for example above a solid surface in a gravitational field. The PE of an object at height h is PE = mgh Units are kg . m . m sec2
An object falling under gravity loses Potential Energy and gains Kinetic Energy. An object falling under gravity loses Potential Energy and gains Kinetic Energy. A pendulum in a vacuum has potential energy PE = mgh at the highest points, and no kinetic energy because it stops A pendulum in a vacuum has kinetic energy KE = 1/2 mass.V2 at the lowest point h = 0, and no potential energy. The two energy extremes are equal
We said earlier “Energy is Conserved” We said earlier “Energy is Conserved” This means KE + PE + Pv + Heat = constant For simple systems involving liquid water without friction heat, at two places 1 and 2 1/2 mV12 + mgh1 + P1v = 1/2 mV22 + mgh2 + P2v If both places are at the same pressure (say both touch the atmosphere) the pressure terms are identical
A tank has an opening h = 1 m below the water level. The opening has area A2 = 0.003 m2 , small compared to the tank with area A1 = 3 m2. Therefore assume V1 ~ 0. A tank has an opening h = 1 m below the water level. The opening has area A2 = 0.003 m2 , small compared to the tank with area A1 = 3 m2. Therefore assume V1 ~ 0. Calculate V2. Method: only PE at 1, KE at 2 mgh1=1/2mV22 V2 = 2gh
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