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Equilibrium and Elasticity • Many bodies, such as bridges, aqueducts, and ladders, are designed so they do not accelerate. • Real materials are not truly rigid. They are elastic and do deform to some extent. • We consider stress and strain to understand the deformation of real bodies. Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Tuesday, October 18, 11 Acts like a spring Tuesday, October 18, 11 Tuesday, October 18, 11 Quiz Hooke’s law describes the force of A. B. C. D. E. gravity. a spring. collisions. tension. none of the above. © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-9 Answer Hooke’s law describes the force of A. B. C. D. E. gravity. a spring. collisions. tension. none of the above. © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-10 Tuesday, October 18, 11 Checking Understanding Which spring has the largest spring constant? © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-23 Answer Which spring has the largest spring constant? A © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-24 Checking Understanding The same spring is stretched or compressed as shown below. In which case does the force exerted by the spring have the largest magnitude? © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-25 Answer The same spring is stretched or compressed as shown below. In which case does the force exerted by the spring have the largest magnitude? E. Not enough information to tell. © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-26 Torque and Static Equilibrium For an extended object to be in equilibrium, the net force and the net torque must be zero. © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-11 Choosing the Pivot Point © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-12 Conditions for equilibrium • First condition: The sum of all the forces is equal to zero: ΣFx = 0 ΣFy = 0 ΣFz = 0 • Second condition: The sum of all torques about any given point is equal to zero. Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Center of gravity • We can treat a body’s weight as though it all acts at a single point— the center of gravity. • If we can ignore the variation of gravity with altitude, the center of gravity is the same as the center of mass. Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Solving Static Equilibrium Problems © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-13 Checking Understanding What does the scale read? A. B. C. D. 500 N 1000 N 2000 N 4000 N © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-14 Answer What does the scale read? A. B. C. D. 500 N 1000 N 2000 N 4000 N © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-15 Balance Line of action Gravity acts at the center of gravity. This force exerts no torque about her toes. Base of support For an object to balance, its center of gravity must reside over its base of support. That way gravity does not exert a torque. © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-18 Stability of a Car © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-19 Tiptoeing Why can’t you stand on tiptoes if your toes are against a wall? © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-20 Tiptoeing Why can’t you stand on tiptoes if your toes are against a wall? Center of gravity has to be over toes – the base of support – to balance. That requires shifting your body slightly forward. But you can’t shift your body forward if your toes are against the wall. © 2010 Pearson Education, Inc. Tuesday, October 18, 11 Slide 8-20 Tensile and compressive stress and strain • Tensile stress = F⊥ /A and tensile strain = Δl/l0. • Compressive stress and compressive strain are defined in a similar way. • Young’s modulus is tensile stress divided by tensile strain, and is given by Y = (F⊥/A)(l0/Δl). Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Some values of elastic moduli Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Tensile stress and strain • In many cases, a body can experience both tensile and compressive stress at the same time Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Bulk stress and strain • Pressure in a fluid is force per unit area: p = F⊥/A. • Bulk stress is pressure change Δp and bulk strain is fractional volume change ΔV/V0. • Bulk modulus is bulk stress divided by bulk strain and is given by B = –Δp/(ΔV/V0). Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Sheer stress and strain • Sheer stress is F||/A and sheer strain is x/h • Sheer modulus is sheer stress divided by sheer strain, and is given by S = (F||/A)(h/x). Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Elasticity and plasticity • Hooke’s law applies up to point a • Table 11.3 shows some approximate breaking stresses. Copyright © 2012 Pearson Education Inc. Tuesday, October 18, 11 Q11.1 Which of the following situations satisfies both the first condition for equilibrium (net force = 0) and the second condition for equilibrium (net torque = 0)? A. an automobile crankshaft turning at an increasing angular speed in the engine of a parked car B. a seagull gliding at a constant angle below the horizontal and at a constant speed C. a thrown baseball that does not rotate as it sails through the air D. more than one of the above E. none of the above © 2012 Pearson Education, Inc. Tuesday, October 18, 11 A11.1 Which of the following situations satisfies both the first condition for equilibrium (net force = 0) and the second condition for equilibrium (net torque = 0)? A. an automobile crankshaft turning at an increasing angular speed in the engine of a parked car B. a seagull gliding at a constant angle below the horizontal and at a constant speed C. a thrown baseball that does not rotate as it sails through the air D. more than one of the above E. none of the above © 2012 Pearson Education, Inc. Tuesday, October 18, 11 Q11.2 A rock is attached to the left end of a uniform meter stick that has the same mass as the rock. How far from the left end of the stick should the triangular object be placed so that the combination of meter stick and rock is in balance? A.less than 0.25 m B. 0.25 m C. between 0.25 m and 0.50 m D. 0.50 m E. more than 0.50 m © 2012 Pearson Education, Inc. Tuesday, October 18, 11 A11.2 A rock is attached to the left end of a uniform meter stick that has the same mass as the rock. How far from the left end of the stick should the triangular object be placed so that the combination of meter stick and rock is in balance? A.less than 0.25 m L B. 0.25 m C. between 0.25 m and 0.50 m D. 0.50 m E. more than 0.50 m © 2012 Pearson Education, Inc. Tuesday, October 18, 11 0.5-L L × m = (0.5 − L) × m L = (0.5 − L) 2L = 0.5 L = 0.25 Q11.3 A metal advertising sign (weight w) is suspended from the end of a massless rod of length L. The rod is supported at one end by a hinge at point P and at the other end by a cable at an angle θ from the horizontal. What is the tension in the cable? A. T = w sin θ B. T = w cos θ C. T = w/(sin θ) D. T = w/(cos θ) E. none of the above © 2012 Pearson Education, Inc. Tuesday, October 18, 11 A11.3 A metal advertising sign (weight w) is suspended from the end of a massless rod of length L. The rod is supported at one end by a hinge at point P and at the other end by a cable at an angle θ from the horizontal. What is the tension in the cable? A. T = w sin θ B. T = w cos θ C. T = w/(sin θ) D. T = w/(cos θ) E. none of the above © 2012 Pearson Education, Inc. Tuesday, October 18, 11 Q11.5 Two rods are made of the same kind of steel and have the same diameter. F F length L length 2L A force of magnitude F is applied to the end of each rod. Compared to the rod of length L, the rod of length 2L has A. more stress and more strain. B. the same stress and more strain. C. the same stress and less strain. D. less stress and less strain. E. the same stress and the same strain. © 2012 Pearson Education, Inc. Tuesday, October 18, 11 F F A11.5 Two rods are made of the same kind of steel and have the same diameter. F F length L length 2L A force of magnitude F is applied to the end of each rod. Compared to the rod of length L, the rod of length 2L has A. more stress and more strain. B. the same stress and more strain. C. the same stress and less strain. D. less stress and less strain. E. the same stress and the same strain. © 2012 Pearson Education, Inc. Tuesday, October 18, 11 F F Q11.6 Two rods are made of the same kind of steel. The longer rod has a greater diameter. F F length L length 2L A force of magnitude F is applied to the end of each rod. Compared to the rod of length L, the rod of length 2L has A. more stress and more strain. B. the same stress and more strain. C. the same stress and less strain. D. less stress and less strain. E. the same stress and the same strain. © 2012 Pearson Education, Inc. Tuesday, October 18, 11 F F A11.6 Two rods are made of the same kind of steel. The longer rod has a greater diameter. F F length L length 2L A force of magnitude F is applied to the end of each rod. Compared to the rod of length L, the rod of length 2L has A. more stress and more strain. B. the same stress and more strain. C. the same stress and less strain. D. less stress and less strain. E. the same stress and the same strain. © 2012 Pearson Education, Inc. Tuesday, October 18, 11 F F Energy and Work Tuesday, October 18, 11 What Energy Transfer is Occurring Here? Tuesday, October 18, 11 Tuesday, October 18, 11 How do we transfer energy? Tuesday, October 18, 11 How do we transfer energy? Tuesday, October 18, 11 How do we transfer energy? Tuesday, October 18, 11 Tuesday, October 18, 11 Tuesday, October 18, 11 Power: The rate at which energy is transformed from one kind to another. Tuesday, October 18, 11 Quiz 1. If a system is isolated, the total energy of the system A. B. C. D. E. Tuesday, October 18, 11 increases constantly. decreases constantly. is constant. depends on work into the system. depends on work out of the system. Answer 1. If a system is isolated, the total energy of the system A. B. C. D. E. Tuesday, October 18, 11 increases constantly. decreases constantly. is constant. depends on work into the system. depends on work out of the system. Quiz • If you raise an object to a greater height, you are increasing A. B. C. D. E. kinetic energy. heat. potential energy. chemical energy. thermal energy. Slide 10-10 Tuesday, October 18, 11 Answer • If you raise an object to a greater height, you are increasing A. B. C. D. E. Tuesday, October 18, 11 kinetic energy. heat. potential energy. chemical energy. thermal energy. Forms of Energy Mechanical Energy Thermal Energy Tuesday, October 18, 11 Other forms include The Basic Energy Model Tuesday, October 18, 11 Energy Transformations Kinetic energy K = energy of motion Potential energy U = energy of position Thermal energy Eth = energy associated with temperature System energy E = K + U + Eth + Echem + ... Energy can be transformed within the system without loss. Energy is a property of a system. Tuesday, October 18, 11 Energy Transformations Kinetic energy K = energy of motion Potential energy U = energy of position Thermal energy Eth = energy associated with temperature System energy E = K + U + Eth + Echem + ... Energy can be transformed within the system without loss. Energy is a property of a system. Tuesday, October 18, 11 Some Energy Transformations Tuesday, October 18, 11 Some Energy Transformations Echem → Ug Tuesday, October 18, 11 Some Energy Transformations Echem → Ug Tuesday, October 18, 11 Some Energy Transformations Echem → Ug Tuesday, October 18, 11 K → Eth Some Energy Transformations Echem → Ug Tuesday, October 18, 11 K → Eth Some Energy Transformations Echem → Ug Echem → Ug Echem → Eth Tuesday, October 18, 11 K → Eth Some Energy Transformations Echem → Ug Echem → Ug Echem → Eth Tuesday, October 18, 11 K → Eth Some Energy Transformations Echem → Ug K → Eth Echem → Ug Us → K → Ug Echem → Eth Tuesday, October 18, 11 Checking Understanding A child is on a playground swing, motionless at the highest point of his arc. As he swings back down to the lowest point of his motion, what energy transformation is taking place? A. K → Ug B. Ug→ Eth C. Us → Ug D. Ug → K E. K → Eth Tuesday, October 18, 11 Answer A child is on a playground swing, motionless at the highest point of his arc. As he swings back down to the lowest point of his motion, what energy transformation is taking place? A. K → Ug B. Ug→ Eth C. Us → Ug D. Ug → K E. K → Eth Tuesday, October 18, 11 Energy Transfers These change the energy of the system. Interactions with the environment. Work is the mechanical transfer of energy to or from a system via pushes and pulls. Tuesday, October 18, 11 Energy Transfers: Work Tuesday, October 18, 11 Energy Transfers: Work W→ K Tuesday, October 18, 11 Energy Transfers: Work W→ K Tuesday, October 18, 11 Energy Transfers: Work W→ K Tuesday, October 18, 11 W→ Eth Energy Transfers: Work W→ K Tuesday, October 18, 11 W→ Eth Energy Transfers: Work W→ K W→ Eth W→ Us Elastic Potential Energy Tuesday, October 18, 11 In part (a) of the figure, an air track cart attached to a spring rests on the track at the position xequilibrium and the spring is relaxed. In (b), the cart is pulled to the position xstart and released. It then oscillates about xequilibrium. Which graph correctly represents the potential energy of the spring as a function of the position of the cart? Tuesday, October 18, 11 In part (a) of the figure, an air track cart attached to a spring rests on the track at the position xequilibrium and the spring is relaxed. In (b), the cart is pulled to the position xstart and released. It then oscillates about xequilibrium. Which graph correctly represents the potential energy of the spring as a function of the position of the cart? Answer: 3.The cart starts at xstart with no kinetic energy, and so the spring’s ✔ potential energy is a maximum. Once released, the cart accelerates to the right and its kinetic energy increases as the potential energy of the spring is converted into kinetic energy of the cart. As the cart passes the equilibrium position, its kinetic energy is a maximum and so the spring’s potential energy is a minimum. Once to the right of xequilibrium, the cart starts to compress the spring and it slows down as its kinetic energy is converted back to potential energy of the recompressed spring. At the rightmost point it reaches, the cart reverses its direction of travel. At that instant, it has no kinetic energy and the spring again has maximum potential energy. Tuesday, October 18, 11 Tuesday, October 18, 11 • We’ve studied how Newton’s Second Law allows us to calculate an acceleration from a force but what if the force changes during its application? We must be able to account for things like an archer’s bow. • We must look at action– reaction pairs that are not immediately obvious (like the shotgun expelling the pellets with expanding gas but having the expanding gas do work on the shotgun at the same Tuesday, October 18, 11 Tuesday, October 18, 11 The Law of Conservation of Energy Tuesday, October 18, 11 The Work-Energy Equation Tuesday, October 18, 11 Work Tuesday, October 18, 11 Work Done by Force at an Angle to Displacement Tuesday, October 18, 11 Use the parallel component if the force acts at an angle Tuesday, October 18, 11 How can it be such a great “workout” with no work? When positive and negative work cancel, the net work is zero even though muscles are exercising. Tuesday, October 18, 11 The work-energy theorem Work done on an object can change its motion and energy. Tuesday, October 18, 11 We can compare the kinetic energy of different bodies Changes in the energy of a moving body under the influence of an applied force change differently depending on the direction of application. Tuesday, October 18, 11 The Basic Equation Kf + Uf + ΔEth = Ki + Ui A few things to note: • Work can be positive (work in) or negative (work out) • We are, for now, ignoring heat. • Thermal energy is…special. When energy changes to thermal energy, this change is irreversible. Tuesday, October 18, 11 Stepwise solution of work done by several forces Tuesday, October 18, 11 Stepwise solution of work done by several forces Tuesday, October 18, 11 Energy Equations Tuesday, October 18, 11 Checking Understanding Each of the boxes, with masses noted, is pulled for 10 m across a level, frictionless floor by the noted force. Which box experiences the largest change in kinetic energy? Tuesday, October 18, 11 Answer Each of the boxes, with masses noted, is pulled for 10 m across a level, frictionless floor by the noted force. Which box experiences the largest change in kinetic energy? D. Tuesday, October 18, 11 Checking Understanding Each of the boxes, with masses noted, is pulled for 10 m across a level, frictionless floor by the noted force. Which box experiences the smallest change in kinetic energy? Slide 10-33 Tuesday, October 18, 11 Answer Each of the boxes, with masses noted, is pulled for 10 m across a level, frictionless floor by the noted force. Which box experiences the smallest change in kinetic energy? C. Tuesday, October 18, 11 Example Problem A 200 g block on a frictionless surface is pushed against a spring with spring constant 500 N/m, compressing the spring by 2.0 cm. When the block is released, at what speed does it shoot away from the spring? Tuesday, October 18, 11 Example Problem A 200 g block on a frictionless surface is pushed against a spring with spring constant 500 N/m, compressing the spring by 2.0 cm. When the block is released, at what speed does it shoot away from the spring? Use : F = ma (Fsp )x = −kΔx v = v + 2ax (x − x0 ) 2 x Tuesday, October 18, 11 2 0x Tuesday, October 18, 11 Work and energy with varying forces Perhaps the best example is driving a car, alternating your attention between the gas and the brake. The effect is a variable positive or negative force of various magnitude along a straight line. Tuesday, October 18, 11 The stretch of a spring and the force that caused it The force applied to an ideal spring will be proportional to its stretch. The graph of force on the y axis versus stretch on the x axis will yield a slope of k, the spring constant. Tuesday, October 18, 11 Stepping on a scale Whether you like the result or not, stepping on a scale is an excellent example of applied force and the work being done to compress that spring. Tuesday, October 18, 11 Motion with a varying force Tuesday, October 18, 11 Motion with a varying force Tuesday, October 18, 11 Motion on a curved path If you watch a child on a swing set, you can also consider the motion of a particle along a curved path. Tuesday, October 18, 11 Watt about power? Once work is calculated, dividing by the time that passed determines power. The pun is credit to James Watt. (You will see that scientists of that era often were privileged to leave their names on the topic of their efforts.) Also note the popular culture power unit of horsepower. The energy you use may be noted from the meter the electric company probably installed to measure your consumption of energy in kilowatt-hours. Tuesday, October 18, 11 Power Tuesday, October 18, 11 Power Tuesday, October 18, 11 Power • • Same mass... Both reach 60 mph... Same final kinetic energy, but different times mean different powers. Tuesday, October 18, 11 An example you might do if the elevator is out It’s interesting how a lighter stair climber and heavier stair climber can expend the same power by using different amounts of time. Tuesday, October 18, 11 Elastic Collisions Tuesday, October 18, 11 Checking Understanding Four toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the greatest power? Tuesday, October 18, 11 Car Mass (g) Speed (m/s) Time (s) A 100 3 2 B 200 2 2 C 200 2 3 D 300 2 3 E 300 1 4 Answer Four toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the greatest power? Tuesday, October 18, 11 Car Mass (g) Speed (m/s) Time (s) A 100 3 2 B 200 2 2 C 200 2 3 D 300 2 3 E 300 1 4 Checking Understanding Four toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the smallest power? Tuesday, October 18, 11 Car Mass (g) Speed (m/s) Time (s) A 100 3 2 B 200 2 2 C 200 2 3 D 300 2 3 E 300 1 4 Answer Four toy cars accelerate from rest to their top speed in a certain amount of time. The masses of the cars, the final speeds, and the time to reach this speed are noted in the table. Which car has the smallest power? Tuesday, October 18, 11 Car Mass (g) Speed (m/s) Time (s) A 100 3 2 B 200 2 2 C 200 2 3 D 300 2 3 E 300 1 4 Summary Tuesday, October 18, 11 Summary Tuesday, October 18, 11 Additional Questions Each of the 1.0 kg boxes starts at rest and is then pushed for 2.0 m across a level, frictionless floor by a rope with the noted force. Which box has the highest final speed? Tuesday, October 18, 11 Answer Each of the 1.0 kg boxes starts at rest and is then pushed for 2.0 m across a level, frictionless floor by a rope with the noted force. Which box has the highest final speed? E. Slide 10-51 Tuesday, October 18, 11