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PHYS 1441 – Section 002 Lecture # 23

PHYS 1441 – Section 002 Lecture # 23. Monday , Dec. 6, 2010 Dr. Jae hoon Yu. Similarities Between Linear and Rotational Quantities Conditions for Equilibrium How to Solve Equilibrium Problems? A Few Examples of Mechanical Equilibrium Elastic Properties of Solids

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PHYS 1441 – Section 002 Lecture # 23

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  1. PHYS 1441 – Section 002Lecture #23 Monday, Dec. 6, 2010 Dr. Jaehoon Yu • Similarities Between Linear and Rotational Quantities • Conditions for Equilibrium • How to Solve Equilibrium Problems? • A Few Examples of Mechanical Equilibrium • Elastic Properties of Solids • Density and Specific Gravity PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  2. Announcements • The Final Exam • Date and time: 11am – 1:30pm, Monday Dec. 13 • Place: SH103 • Comprehensive exam • Covers from CH1.1 – what we finish Wednesday, Dec. 8 • Plus appendices A.1 – A.8 • Combination of multiple choice and free response problems • Bring your Planetarium extra credit sheet to the class next Wednesday, Dec. 8, with your name clearly marked on the sheet! PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  3. Similarity Between Linear and Rotational Motions All physical quantities in linear and rotational motions show striking similarity. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  4. F d CM d -F Conditions for Equilibrium What do you think the term “An object is at its equilibrium” means? The object is either at rest (Static Equilibrium) or its center of mass is moving at a constant velocity (Dynamic Equilibrium). When do you think an object is at its equilibrium? Translational Equilibrium: Equilibrium in linear motion Is this it? The above condition is sufficient for a point-like object to be at its translational equilibrium. However for an object with size this is not sufficient. One more condition is needed. What is it? Let’s consider two forces equal in magnitude but in opposite direction acting on a rigid object as shown in the figure. What do you think will happen? The object will rotate about the CM. Since the net torque acting on the object about a rotational axis is not0. For an object to be at its static equilibrium, the object should not have linear or angular speed. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  5. r’ F1 O O’ r5 F3 F2 F4 F5 More on Conditions for Equilibrium To simplify the problem, we will only deal with forces acting on x-y plane, giving torque only along z-axis. What do you think the conditions for equilibrium be in this case? The six possible equations from the two vector equations turns to three equations. AND What happens if there are many forces exerting on an object? If an object is at its translational static equilibrium, and if the net torque acting on the object is 0 about one axis, the net torque must be 0 about any arbitrary axis. Why is this true? Because the object is not moving, no matter what the rotational axis is, there should not be any motion. It is simply a matter of mathematical manipulation. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  6. How do we solve static equilibrium problems? • Select the object to which the equations for equilibrium are to be applied. • Identify all the forces and draw a free-body diagram with them indicated on it with their directions and locations properlyindicated • Choose a convenient set of x and y axes and write down force equation for each x and y component with correct signs. • Apply the equations that specify the balance of forces at equilibrium. Set the net force in the x and y directions equal to 0. • Select the most optimal rotational axis for torque calculations  Selecting the axis such that the torque of one or more of the unknown forces become 0 makes the problem much easier to solve. • Write down the torque equation with proper signs. • Solve the force and torque equations for the desired unknown quantities. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  7. 1m x n F D MBg MDg MFg Example for Mechanical Equilibrium A uniform 40.0 N board supports the father and the daughter each weighing 800 N and 350 N, respectively, and is not moving. If the support (or fulcrum) is under the center of gravity of the board, and the father is 1.00 m from CoG, what is the magnitude of the normal force n exerted on the board by the support? Since there is no linear motion, this system is in its translational equilibrium Therefore the magnitude of the normal force Determine where the child should sit to balance the system. The net torque about the fulcrum by the three forces are Therefore to balance the system the daughter must sit PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  8. Rotational axis x/2 F D 1m x n MFg MFg MBg Example for Mech. Equilibrium Cont’d Determine the position of the child to balance the system for different position of axis of rotation. The net torque about the axis of rotation by all the forces are Since the normal force is The net torque can be rewritten What do we learn? Therefore No matter where the rotation axis is, net effect of the torque is identical. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  9. FB d O l mg FU Ex.9.8for Mechanical Equilibrium A person holds a 50.0N sphere in his hand. The forearm is horizontal. The biceps muscle is attached 3.00 cm from the joint, and the sphere is 35.0cm from the joint. Find the upward force exerted by the biceps on the forearm and the downward force exerted by the upper arm on the forearm and acting at the joint. Neglect the weight of forearm. Since the system is in equilibrium, from the translational equilibrium condition From the rotational equilibrium condition Thus, the force exerted by the biceps muscle is Force exerted by the upper arm is PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  10. Elastic Properties of Solids We have been assuming that the objects do not change their shapes when external forces are exerting on it. It this realistic? No. In reality, the objects get deformed as external forces act on it, though the internal forces resist the deformation as it takes place. Deformation of solids can be understood in terms of Stress and Strain Stress: A quantity proportional to the force causing the deformation. Strain: Measure of the degree of deformation It is empirically known that for small stresses, strain is proportional to stress The constants of proportionality are called Elastic Modulus • Young’s modulus: Measure of the elasticity in a length • Shear modulus: Measure of the elasticity in an area • Bulk modulus: Measure of the elasticity in a volume Three types of Elastic Modulus PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  11. FW mg FGy O FGx Example 9 – 7 A 5.0 m long ladder leans against a wall at a point 4.0m above the ground. The ladder is uniform and has mass 12.0kg. Assuming the wall is frictionless (but ground is not), determine the forces exerted on the ladder by the ground and the wall. First the translational equilibrium, using components FBD Thus, the y component of the force by the ground is The length x0 is, from Pythagorian theorem PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  12. Example 9 – 7 cont’d From the rotational equilibrium Thus the force exerted on the ladder by the wall is The x component of the force by the ground is Solve for FGx Thus the force exerted on the ladder by the ground is The angle between the ground force to the floor PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  13. Ex. A Diving Board A woman whose weight is 530 N is poised at the right end of a diving boardwith length 3.90 m. The board hasnegligible weight and is supported bya fulcrum 1.40 m away from the leftend.Find the forces that the bolt and the fulcrum exert on the board. How large is the torque by the bolt? First the torque eq. None Why? So the force by the fulcrum is Because the lever arm is 0. Now the force eq. So the force by the bolt is PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  14. pinA mg pAA Finger Holds Water in Straw You insert a straw of length L into a tall glass of your favorite beverage. You place your finger over the top of the straw so that no air can get in or out, and then lift the straw from the liquid. You find that the straw strains the liquid such that the distance from the bottom of your finger to the top of the liquid is h. Does the air in the space between your finger and the top of the liquid have a pressure P that is (a) greater than, (b) equal to, or (c) less than, the atmospheric pressure PA outside the straw? Less What are the forces in this problem? Gravitational force on the mass of the liquid Force exerted on the top surface of the liquid by inside air pressure Force exerted on the bottom surface of the liquid by the outside air Since it is at equilibrium Cancel A and solve for pin So pin is less than PA by rg(L-h). PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  15. Li Lf=Li+DL Fex Fex Fin A:cross sectional area Young’s Modulus Let’s consider a long bar with cross sectional area A and initial length Li. After the stretch Fex=Fin Tensile stress Tensile strain Used to characterize a rod or wire stressed under tension or compression Young’s Modulus is defined as What is the unit of Young’s Modulus? Force per unit area • For a fixed external force, the change in length is proportional to the original length • The necessary force to produce the given strain is proportional to the cross sectional area Experimental Observations Elastic limit: Maximum stress that can be applied to the substance before it becomes permanently deformed PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  16. F V’ F F F Because the change of volume is reverse to change of pressure. Bulk Modulus Bulk Modulus characterizes the response of a substance to uniform squeezing or reduction of pressure. After the pressure change V Volume stress =pressure If the pressure on an object changes by DP=DF/A, the object will undergo a volume change DV. Bulk Modulus is defined as Compressibility is the reciprocal of Bulk Modulus PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  17. Example for Solid’s Elastic Property A solid brass sphere is initially under normal atmospheric pressure of 1.0x105N/m2. The sphere is lowered into the ocean to a depth at which the pressures is 2.0x107N/m2. The volume of the sphere in air is 0.5m3. By how much its volume change once the sphere is submerged? Since bulk modulus is The amount of volume change is From table 12.1, bulk modulus of brass is 6.1x1010 N/m2 The pressure change DP is Therefore the resulting volume change DV is The volume has decreased. PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  18. Density and Specific Gravity Density, r (rho), of an object is defined as mass per unit volume Unit? Dimension? Specific Gravity of a substance is defined as the ratio of the density of the substance to that of water at 4.0 oC (rH2O=1.00g/cm3). Unit? None Dimension? None Sink in the water What do you think would happen of a substance in the water dependent on SG? Float on the surface PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  19. Fluid and Pressure What are the three states of matter? Solid, Liquid and Gas Using the time it takes for a particular substance to change its shape in reaction to external forces. How do you distinguish them? A collection of molecules that are randomly arranged and loosely bound by forces between them or by an external container. What is a fluid? We will first learn about mechanics of fluid at rest, fluid statics. In what ways do you think fluid exerts stress on the object submerged in it? Fluid cannot exert shearing or tensile stress. Thus, the only force the fluid exerts on an object immersed in it is the force perpendicular to the surface of the object. This force by the fluid on an object usually is expressed in the form of the force per unit area at the given depth, the pressure, defined as Expression of pressure for an infinitesimal area dA by the force dF is Note that pressure is a scalar quantity because it’s the magnitude of the force on a surface area A. Special SI unit for pressure is Pascal What is the unit and the dimension of pressure? Unit:N/m2 Dim.: [M][L-1][T-2] PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  20. Example for Pressure The mattress of a water bed is 2.00m long by 2.00m wide and 30.0cm deep. a) Find the weight of the water in the mattress. The volume density of water at the normal condition (0oC and 1 atm) is 1000kg/m3. So the total mass of the water in the mattress is Therefore the weight of the water in the mattress is b) Find the pressure exerted by the water on the floor when the bed rests in its normal position, assuming the entire lower surface of the mattress makes contact with the floor. Since the surface area of the mattress is 4.00 m2, the pressure exerted on the floor is PHYS 1441-002, Fall 2009 Dr. Jaehoon Yu

  21. h Mg B Buoyant Forces and Archimedes’ Principle Why is it so hard to put an inflated beach ball under water while a small piece of steel sinks in the water easily? The water exerts force on an object immersed in the water. This force is called the buoyant force. How large is the buoyant force? The magnitude of the buoyant force always equals the weight of the fluid in the volume displaced by the submerged object. This is called Archimedes’ principle. What does this mean? Let‘s consider a cube whose height is h and is filled with fluid and in its equilibrium so that its weight Mg is balanced by the buoyant force B. The pressure at the bottom of the cube is larger than the top by rgh. Therefore, Where Mg is the weight of the fluid in the cube. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  22. h Mg B More Archimedes’ Principle Let’s consider the buoyant force in two special cases. Case 1: Totally submerged object Let’s consider an object of mass M, with density r0, is fully immersed in the fluid with density rf . The magnitude of the buoyant force is The weight of the object is Therefore total force in the system is • The total force applies to different directions depending on the difference of the density between the object and the fluid. • If the density of the object is smaller than the density of the fluid, the buoyant force will push the object up to the surface. • If the density of the object is larger than the fluid’s, the object will sink to the bottom of the fluid. What does this tell you? PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  23. h Mg B More Archimedes’ Principle Case 2: Floating object Let’s consider an object of mass M, with density r0, is in static equilibrium floating on the surface of the fluid with density rf, and the volume submerged in the fluid is Vf. The magnitude of the buoyant force is The weight of the object is Therefore total force of the system is Since the system is in static equilibrium Since the object is floating, its density is smaller than that of the fluid. The ratio of the densities between the fluid and the object determines the submerged volume under the surface. What does this tell you? PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  24. Example for Archimedes’ Principle Archimedes was asked to determine the purity of the gold used in the crown. The legend says that he solved this problem by weighing the crown in air and in water. Suppose the scale read 7.84N in air and 6.86N in water. What should he have to tell the king about the purity of the gold in the crown? In the air the tension exerted by the scale on the object is the weight of the crown In the water the tension exerted by the scale on the object is Therefore the buoyant force B is Since the buoyant force B is The volume of the displaced water by the crown is Therefore the density of the crown is Since the density of pure gold is 19.3x103kg/m3, this crown is not made of pure gold. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  25. Example for Buoyant Force What fraction of an iceberg is submerged in the sea water? Let’s assume that the total volume of the iceberg is Vi. Then the weight of the iceberg Fgi is Let’s then assume that the volume of the iceberg submerged in the sea water is Vw. The buoyant force B caused by the displaced water becomes Since the whole system is at its static equilibrium, we obtain Therefore the fraction of the volume of the iceberg submerged under the surface of the sea water is About 90% of the entire iceberg is submerged in the water!!! PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  26. Flow Rate and the Equation of Continuity Study of fluid in motion: Fluid Dynamics If the fluid is water: Hydro-dynamics Water dynamics?? • 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 absorbs a lot of energy Two main types of flow Flow rate: the mass of fluid that passes the given point per unit time since the total flow must be conserved Equation of Continuity PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  27. Example for Equation of Continuity How large must a heating duct be if air moving at 3.0m/s through it can replenish the air in a room of 300m3 volume every 15 minutes? Assume the air’s density remains constant. Using equation of continuity Since the air density is constant Now let’s imagine the room as the large section of the duct PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  28. Bernoulli’s Principle 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 the work done by the force, F1, that exerts pressure, P1, at point 1 Amount of the work done by the force in the other section of the fluid is Work done by the gravitational force to move the fluid mass, m, from y1 to y2 is PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  29. Bernoulli’s Equation cont’d The total amount of the work done on the fluid is From the work-energy principle Since the mass m is contained in the volume that flowed in the motion and Thus, PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  30. Since Bernoulli’s Equation cont’d We obtain Re-organize Bernoulli’s Equation Thus, for any two points in the flow Result of Energy conservation! Pascal’s Law For static fluid For the same heights The pressure at the faster section of the fluid is smaller than slower section. PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

  31. Example for Bernoulli’s Equation Water circulates throughout a house in a hot-water heating system. If the water is pumped at the 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 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 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu

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