Monday April 26, PHYS , Spring 2004 Dr. Andrew Brandt PHYS 1443 – Section 501 Lecture #24 Monday, April 26, 2004 Dr. Andrew Brandt 1.Fluid Dymanics : Flow rate and Continuity Equation 2.Bernoulli’s Equation 3.Simple Harmonic Motion 4.Simple Block-Spring System

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 2 Announcements The last HW on Ch is due Weds. 4/28 (decided not to have HW during last week) An optional take-home quiz to replace your lowest quiz is due tonight No optional sections (*) in Ch 11, 12, 13 Explicitly, covered , We’ll stop class a little early for teaching evaluations

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 3 Pascal’s Law and Hydraulics A change in the pressure applied to a fluid is transmitted undiminished to every point of the fluid and to the walls of the container. The resultant pressure P at any given depth h increases as much as the change in P 0. This is the principle behind hydraulic pressure. Therefore, the resultant force F 2 is What happens if P 0 is changed? Since the pressure change caused by the force F1 F1 applied on to the area A1 A1 is transmitted to the F2 F2 on an area A2.A2. d1d1 d2d2 F1F1 A1A1 A2A2 F2F2 In other words, the initial force multiplied by the ratio of the areas A 2 /A 1 is transmitted to F2 F2 on the surface. Note the actual displaced volume of the fluid is the same and the work done by the forces are still the same.

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 4 Buoyant Forces and Archimedes’ Principle Why is it so hard to put a beach ball under water while a piece of small steel sinks in the water? The water exerts a force on an object immersed in the water. This force is called Buoyant force. How does the Buoyant force work? Let’s consider a cube whose height is h and is filled with fluid and at its equilibrium. Then the weight Mg is balanced by the buoyant force B. This is called Archimedes’ principle. The magnitude of the buoyant force always equals the weight of the volume of fluid displaced by the submerged object. B MgMg h And the pressure at the bottom of the cube is larger than the top by  gh. Therefore, Where Mg is the weight of the fluid.

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 5 Flow Rate and the Equation of Continuity Study of fluid in motion: Fluid Dynamics If the fluid is water: Streamline or Laminar flow : Each particle of the fluid follows a smooth path, a streamline Turbulent flow : Erratic, small, whirlpool-like circles called eddy current or eddies which absorb a lot of energy Two main types of flow Hydro-dynamics Flow rate: the mass of fluid that passes a given point per unit time since the total flow must be conserved Equation of Continuity

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 6 Example for Equation of Continuity How large must a heating duct be if air moving at 3.0m/s along it can replenish the air every 15 minutes in a room of 300m 3 volume? Assume the air’s density remains constant. Using equation of continuity Since the air density is constant Now let’s view the room as a large section of the duct

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 7 Bernoulli’s Equation Bernoulli’s Principle: Where the velocity of fluid is high, the pressure is low, and where the velocity is low, the pressure is high. Amount of work done by the force, F1,F1, that exerts pressure, P 1, at point 1 Work done by the gravitational force to move the fluid mass, m, from y1 y1 to y2 y2 is Amount of work done on the other section of the fluid is

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 8 Bernoulli’s Equation cont’d The net work done on the fluid is From the work-energy principle Since mass, m, is contained in the volume that flowed in the motion and Thus,

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 9 Bernoulli’s Equation cont’d We obtain Re- organize Bernoulli’s Equation Since Thus, for any two points in the flow For static fluid For the same heights The pressure at the faster section of the fluid is smaller than slower section. Pascal’s Law

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 10 Example for Bernoulli’s Equation Water circulates throughout a house in a hot-water heating system. If the water is pumped at a speed of 0.5m/s through a 4.0cm diameter pipe in the basement under a pressure of 3.0atm, what will be the flow speed and pressure in a 2.6cm diameter pipe on the second floor 5.0m above? Assume the pipes do not divide into branches. Using the equation of continuity, flow speed on the second floor is Using Bernoulli’s equation, the pressure in the pipe on the second floor is

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 11 Simple Harmonic Motion Harmonic Motion is motion that occurs due to a force that depends on displacement, with the force always directed toward the system’s equilibrium position. When a spring is stretched from its equilibrium position by a length x, the force acting on the mass is What is a system that has such characteristics? A system consists of a mass and a spring This is a second order differential equation that can be solved but it is beyond the scope of this class. It’s negative, because the force resists against the change of length, and is thus directed toward the equilibrium position. From Newton’s second law we obtain What do you observe from this equation? Acceleration is proportional to displacement from the equilibrium Acceleration is opposite direction to displacement Condition for simple harmonic motion

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 12 Equation of Simple Harmonic Motion The solution for the 2 nd order differential equation What happens when t=0 and  =0? Let’s think about the meaning of this equation of motion What are the maximum/minimum possible values of x? Amplitude Phase Angular Frequency Phase constant What is general expression for  ? A/-A An oscillation is fully characterized by its: Amplitude Period or frequency Phase constant Generalized expression of a simple harmonic motion

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 13 More on Equation of Simple Harmonic Motion Let’s now think about the object’s speed and acceleration. Since after a full cycle the position must be the same Speed at any given time The period What is the time for full cycle of oscillation? One of the properties of an oscillatory motion Frequency How many full cycles of oscillation does this undergo per unit time? What is the unit? 1/s=Hz Max speed Max acceleration Acceleration at any given time What do we learn about acceleration? Acceleration is in opposite direction of displacement Acceleration and speed are  /2 out of phase

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 14 Simple Harmonic Motion continued Let’s determine the phase constant and amplitude By taking the ratio, one can obtain the phase constant Phase constant determines the starting position of a simple harmonic motion. This constant is important when there are more than one harmonic oscillation involved in the motion to determine the overall effect of the composite motion At t=0 By squaring the two equation and adding them together, one can obtain the amplitude

Monday April 26, 2004PHYS , Spring 2004 Dr. Andrew Brandt 15 Example for Simple Harmonic Motion From the equation of motion: Taking the first derivative on the equation of motion, the velocity is An object oscillates with simple harmonic motion along the x-axis. Its displacement from the origin varies with time according to the equation; where t is in seconds and the angles is in the parentheses are in radians. a) Determine the amplitude, frequency, and period of the motion. The amplitude, A, is The angular frequency, , is Therefore, frequency and period are b)Calculate the velocity and acceleration of the object at any time t. By the same token, taking the second derivative of equation of motion, the acceleration, a, is