PHYS 1443 Section 003 Lecture #21 Wednesday, Nov. 19, 2003 Dr. Mystery Lecturer

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1 PHYS 443 Section 003 Lecture # Wednesday, Nov. 9, 003 Dr. Mystery Lecturer. Fluid Dyanics : Flow rate and Continuity Equation. Bernoulli s Equation 3. Siple Haronic Motion 4. Siple Bloc-Spring Syste 5. Energy of the Siple Haronic Oscillator Today s Hoewor is # due on Wednesday, Nov. 6, 003!! Net Wednesday s class is cancelled but there will be hoewor!! Wednesday, Nov. 9, 003 PHYS , Fall 003

Flow Rate and the Equation of Continuity Study of fluid in otion: Fluid Dynaics If the fluid is water: Two ain types of flow Water Hydro-dynaics dynaics?

2 Flow Rate and the Equation of Continuity Study of fluid in otion: Fluid Dynaics If the fluid is water: Two ain types of flow Water Hydro-dynaics dynaics?? Strealine or Lainar flow: Each particle of the fluid follows a sooth path, a strealine Turbulent flow: Erratic, sall, whirlpool-lie circles called eddy current or eddies which absorbs a lot of energy Flow rate: the ass of fluid that passes a given point per unit tie / Wednesday, Nov. 9, 003 PHYS , Fall 003 ρ V ρ A l ρav t t t since the total flow ust be conserved ρ Av ρ Av t t Equation of Continuity t

Eaple for Equation of Continuity How large ust a heating duct be if air oving at 3.0/s along it can replenish the air every 5 inutes in a roo of 300 3 volue? Assue the air s density reains constant.

3 Eaple for Equation of Continuity How large ust a heating duct be if air oving at 3.0/s along it can replenish the air every 5 inutes in a roo of volue? Assue the air s density reains constant. A Av v Al v Wednesday, Nov. 9, 003 PHYS , Fall 003 / t Using equation of continuity ρ Av ρ Av Since the air density is constant Av Av Now let s call the roo as the large section of the duct V v t

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.

4 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. Aount of wor done by the force, F, that eerts pressure, P, at point W F l PA l Aount of wor done on the other section of the fluid is W PA l Wor done by the gravitational force to ove the fluid ass,, fro y to y is ( ) W g y y 3 Wednesday, Nov. 9, 003 PHYS , Fall 003 4

5 Bernoulli s Equation cont d The net wor done on the fluid is W W + W+ W3 PA l PA l gy + gy Fro the wor-energy principle v v Since ass,, is contained in the volue that flowed in the otion Thus, A l A l and ρa l ρa l ρa l v ρa l v PA l PA l gy + gy PA l P A l ρa l gy + ρa l gy Wednesday, Nov. 9, 003 PHYS , Fall 003 5

6 Since We obtain Reorganize Bernoulli s Equation cont d ρa l v ρa l v PA l P A l ρa l gy + ρa l gy ρv ρv P P ρgy + ρgy P + ρv + ρgy P + ρv + ρgy Thus, for any two points in the flow P + ρv + ρgy const. For static fluid ρ ( ) P P+ g y y P+ ρgh P P+ ρ v v For the sae heights ( ) Wednesday, Nov. 9, 003 PHYS , Fall 003 Bernoulli s Equation Pascal s Law The pressure at the faster section of the fluid is saller than slower section. 6

7 Eaple for Bernoulli s Equation Water circulates throughout a house in a hot-water heating syste. If the water is puped at a speed of 0.5/s through a 4.0c diaeter pipe in the baseent under a pressure of 3.0at, what will be the flow speed and pressure in a.6c diaeter pipe on the second 5.0 above? Assue the pipes do not divide into branches. Using the equation of continuity, flow speed on the second floor is v Av A π π r v r Using Bernoulli s equation, the pressure in the pipe on the second floor is P P + ρ( v v ) + ρg( y y) N / Wednesday, Nov. 9, 003 PHYS , Fall / s 0.03 ( ) ( ) 7

8 Siple Haronic Motion What do you thin a haronic otion is? Motion that occurs by the force that depends on displaceent, and the force is always directed toward the syste s equilibriu position. What is a syste that has such characteristics? When a spring is stretched fro its equilibriu position by a length, the force acting on the ass is This is a second order differential equation that can be solved but it is beyond the scope of this class. A syste consists of a ass and a spring It s negative, because the force resists against the change of length, directed toward the equilibriu position. F Fro Newton s second law F a we obtain a What do you observe fro this equation? d Acceleration is proportional to displaceent fro the equilibriu Acceleration is opposite direction to displaceent Condition for siple haronic otion Wednesday, Nov. 9, 003 PHYS , Fall 003 8

9 Equation of Siple Haronic Motion The solution for the nd order differential equation Acos( t+ φ) Aplitude Phase Angular Frequency Let s thin about the eaning of this equation of otion What happens when t0 and φ0? What is φ if is not A at t0? What are the aiu/iniu possible values of? Phase constant cos ( 0 + 0) A cos( φ ) ' φ cos ( ' ) A/-A d A A Generalized epression of a siple haronic otion An oscillation is fully characterized by its: Aplitude Period or frequency Phase constant Wednesday, Nov. 9, 003 PHYS , Fall 003 9

10 More on Equation of Siple Haronic Motion What is the tie for full cycle of oscillation? The period Since after a full cycle the position ust be the sae t + T + φ A cos t + π + φ Let s now thin about the object s speed and acceleration. Speed at any given tie Wednesday, Nov. 9, 003 PHYS , Fall 003 T How any full cycles of oscillation does this undergo per unit tie? v Acceleration at any given tie What do we learn about acceleration? A cos ( ( ) ) ( ) π d a One of the properties of an oscillatory otion f T π A sin( t + φ ) dv A cos t + φ ( ) Frequency Ma speed What is the unit? v a /shz Acos( t + φ ) Ma acceleration a a Acceleration is reverse direction to displaceent Acceleration and speed are π/ off phase: When v is aiu, a is at its iniu A A 0

11 Siple Haronic Motion continued Phase constant deterines the starting position of a siple haronic otion. Acos( t + φ ) This constant is iportant when there are ore than one haronic oscillation involved in the otion and to deterine the overall effect of the coposite otion Let s deterine phase constant and aplitude At t0 t 0 Acosφ At t0 i Acosφ By taing the ratio, one can obtain the phase constant By squaring the two equation and adding the together, one can obtain the aplitude vi A sin φ i φ A v i tan A cos sin φ φ vi i A ( cos φ + sin φ ) A i + vi A i vi + Wednesday, Nov. 9, 003 PHYS , Fall 003

12 Eaple for Siple Haronic Motion An object oscillates with siple haronic otion along the -ais. Its displaceent fro π the origin varies with tie according to the equation; ( 4.00) cos πt + where t is in seconds 4 and the angles is in the parentheses are in radians. a) Deterine the aplitude, frequency, and period of the otion. π Fro the equation of otion: Acos ( t +φ ) ( 4.00) cos πt + 4 The aplitude, A, is A The angular frequency,, is π Therefore, frequency π and period are T s f b)calculate the velocity and acceleration of the object at any tie t. Taing the first derivative on the equation of otion, the velocity is By the sae toen, taing the second derivative of equation of otion, the acceleration, a, is π π d v d a Wednesday, Nov. 9, 003 PHYS , Fall 003 T π π s π π 4 4 (.00 π ) sin πt + / s π ( 4.00 π ) cos πt + / s 4

13 Siple Bloc-Spring Syste A bloc attached at the end of a spring on a frictionless surface eperiences acceleration when the spring is displaced fro an equilibriu position. This becoes a second order differential equation d Does this solution satisfy the differential equation? Let s tae derivatives with respect to tie The resulting differential equation becoes Since this satisfies condition for siple haronic otion, we can tae the solution Now the second order derivative becoes d Acos t ( + φ ) a Whenever the force acting on a particle is linearly proportional to the displaceent fro soe equilibriu position and is in the opposite direction, the particle oves in siple haronic otion. Wednesday, Nov. 9, 003 PHYS , Fall d If we denote d d t d ( sin ( φ )) Α cos ( t + φ ) Α t + A ( cos ( + φ )) Α sin ( t + φ )

14 Special case # More Siple Bloc-Spring Syste How do the period and frequency of this haronic otion loo? Since the angular frequency is The period, T, becoes So the frequency is Α cos t f T T π π What can we learn fro these? Frequency and period do not depend on aplitude Period is inversely proportional to spring constant and proportional to ass Let s consider that the spring is stretched to distance A and th e bloc is let go fro rest, giving 0 initial speed; i A, v i 0, d v a d Α sin t a cost Α Α i π π This equation of otion satisfies all the conditions. So it is the solution for this otion. A/ φ Special case # Suppose bloc is given non-zero initial velocity v i to positive at the instant it is at the equilibriu, i 0 Is this a good v i tan tan ( ) π π Α cos t Asin ( t) solution? i Wednesday, Nov. 9, 003 PHYS , Fall 003 4

PHYS 1443 Section 003 Lecture #22

PHYS 1443 Section 003 Lecture #22 PHYS 443 Section 003 Lecture # Monda, Nov. 4, 003. Siple Bloc-Spring Sste. Energ of the Siple Haronic Oscillator 3. Pendulu Siple Pendulu Phsical Pendulu orsion Pendulu 4. Siple Haronic Motion and Unifor