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What do Machines do? Do they allow one to do more work? Not really, at best they make completing a task easier. So then what do Machines do? • Multiply the force. • Multiply the distance. • Change the direction of the force. Work = Force x Distance an object moves while the force is applied. W=Fxd In SI Units: Force is measured in newtons (N) distance is measured in meters (m) Work in N.m which is a joule (J). Named after James Prescott Joule What does work do? Work causes a change in Energy. In other words, it can do any of the following: • Make something move faster. • Lift something up. • Move something against friction. • A combination of the above. Examples of Work: A cart is pushed to the right as illustrated. 50.0 N distance cart is moved 4.00 m. How much work is done on the cart? W = F x d = (50.0 N)(4.00 m) = 200. J Examples of Work: W=Fxd W = (80.0 N)(20. m) W = 1600 J 80.0 N How much work is done on the box? distance box is moved 20. m. A box is lifted as illustrated. Now let’s apply this to some of our machines. The simplest is most likely levers. Class 1 Lever: load on one side of the fulcrum and the effort on the other side. Effort The Fulcrum is the pivot point. The Load is what we are trying to lift or the output of the machine. The Effort is the force that is applied to lift the load or the input of the machine. Load Fulcrum Class 1 Lever: Load on one side of the fulcrum and the Effort on the other side. Effort Load Fulcrum Class 1 Lever: More terminology Note that in lifting the load the Effort moved much farther than the Load. Effort With a smaller Effort we could lift a Load that is heavier. Load Fulcrum Class 1 Lever: More terminology Fulcrum Load = 900N Effort distance, dE = 60 cm Effort = 300 N The Effort moved 60 cm while the Load moved only 20 cm. We moved 3 times farther than the LOAD. Load distance, dL = 20 cm With an Effort of 300 N we were able to lift a Load of 900 N. We multiplied the input force by 3. Class 1 Lever: More terminology Work, W = F x d WIN = E x dE = (300 N)(0.60 m) = 180 J Fulcrum Load = 900N Note: WorkIN = WorkOUT We didn’t do more work, we just did it with less Effort than if I tried to lift it without the lever. Load distance, dL = 20 cm Effort distance, dE = 60 cm WOUT = L x dL = (900 N)(0.20 m) = 180 J Effort = 300 N Class 1 Lever: More terminology MA = Load/Effort MA = (900 N)/(300 N) MA = 3 MA = dE/dL MA = (60 cm)/(20 cm) MA = 3 Fulcrum Load = 900N Effort distance, dE = 60 cm Effort = 300 N Load distance, dL = 20 cm We say that we have a Mechanical Advantage, MA. We can lift 3 times more than our input Effort. Class 1 Lever: More terminology MA = Load/Effort MA = (900 N)/(300 N) MA = 3 MA = dE/dL MA = (60 cm)/(20 cm) MA = 3 Fulcrum Load = 900N Effort distance, dE = 60 cm Effort = 300 N Load distance, dL = 20 cm We triple our Effort (input force) at the expense of moving the Load ⅓ as much. Class 1 Lever: More terminology We can also analyze the lever by measuring the distance from the Effort to the fulcrum (pivot point) and the distance from the Load to the fulcrum. This is called the lever arm or just arm and is often given the variable name “x.” Effort arm, xE = 3.0 m Load arm, xL = 1.0 m Fulcrum Class 1 Lever: More terminology This can also be used to calculate the Mechanical Advantage. MA = xE/xL = (3.00 m)/(1.00 m) = 3 Looks familiar doesn’t it. Effort arm, xE = 3.0 m Load arm, xL = 1.0 m Fulcrum More terminology: Often we use the terms, Ideal Mechanical Advantage, IMA and Actual Mechanical Advantage, AMA IMA = xE/xL or dE/dL AMA = L/E with the load being just what you ultimately wanted to move, excluding anything else that may have to be moved with it. This will become clearer when we look at a 2nd class lever. More terminology: Often we use the terms, Ideal Mechanical Advantage, IMA and Actual Mechanical Advantage, AMA We want to find out how well the particular machine does its work. This is called Efficiency, Eff. Efficiency, Eff = (AMA/IMA) x 100% 2nd class lever: Notice that the Effort and the Load are on the same side of the Fulcrum and the Load is between the Effort and the Fulcrum. Effort Load Fulcrum 2nd class lever: Again the Effort moves much farther than the Load. We are getting more force out than what we put in, but the load only moves a short distance. We are also lifting the lever along with the load. Fulcrum 2nd class lever: IMA = xE/xL = (2.8 m)/(0.35 m) = 8 The load is 900 N and since the Effort has to lift the Load and the lever, let’s say that the Effort is 150 N. Effort Effort arm, xE = 2.8 m Load arm, xL = 0.35 m Load Fulcrum 2nd class lever: IMA = xE/xL = (2.8 m)/(0.35 m) = 8 AMA = L/E = (900 N)/(150 N) = 6 Eff = AMA/IMA = 6/8 x 100% = 75% E = 150N L = 900 N Effort arm, xE = 2.8 m Load arm, xL = 0.35 m Fulcrum 3rd class lever: Notice that the Effort and the Load are on the same side of the Fulcrum and the Effort is between the Load and the Fulcrum. Load Effort Fulcrum 3rd class lever: IMA = xE/xL = (0.50 m)/(2.0 m) = ¼ = 0.25 AMA = L/E = (200 N)/(1000 N) = ⅕ = 0.2 Eff = AMA/IMA = (0.2/0.25) x 100% = 80% L = 200 N E = 1000 N xL = 2.0 m xE = 50. cm Fulcrum There is a lab part of the competition. Let’s look at some of the basic concepts for a lever that is in static equilibrium. The easiest lever to analyze is the first class lever (seesaw), that is balanced by itself. The center of gravity of the lever is on the fulcrum. c.g. If a lever is not moving (rotating) then it is said to be at static equilibrium. When an object is at static equilibrium the following is true: ΣF = 0, that is netF = 0, no unbalanced forces. Στ = 0, that is there are no unbalanced torques. If you place a seesaw so that its center of gravity is on the fulcrum, it will balance. That is, the left side balances the right side. Torque is the tendency of a force to cause an object to rotate around an axis. In the case of a lever, the axis is the fulcrum. Force In this case the force would make the left side of the lever go down or rotate the lever counterclockwise, ccw. Torque is the tendency of a force to cause an object to rotate around an axis. In the case of a lever, the axis is the fulcrum. Force In this case the force would make the left side of the lever go down or rotate the lever counterclockwise, CCW. What if the force is 24 N, what torque is applied? Earlier we talked about the lever arm or arm being the distance from the fulcrum (axis) to the force. We will use the letter “x” as the symbol for lever arm. F = 24 N The symbol for torque is the Greek letter tau, τ x = 1.2 m τ = (F)(x) = (24 N)(1.2 m) τ = 28.8 N.m = 29 N.m A torque of 29 N.m will rotate the lever CCW. The weight of the seesaw on the left creates a torque that tries to rotate the seesaw counter-clockwise, CCW, so that the left side would go down. The weight of the seesaw on the right creates a torque that tries to make it rotate clockwise,CW, so that the right side would go down. F F The two balance each other and it does not rotate. Another way to look at this is that we can place all the weight of the seesaw ( FL ) at its center of gravity. The center of gravity of the seesaw is at the axis of rotation (fulcrum) so the value of the lever arm is zero and the force creates no torque. c.g. Note: The center of gravity may not be at the geometric center. FL Especially when using wooden meter sticks! Sample problem: Two identical 40.0 kg twin girls are sitting on opposite ends of a seesaw that is 4.0 m long and weighs 700 N, so that the center of gravity of the seesaw is on the fulcrum. How do we analyze this situation? c.g. First, we need to draw a torque diagram of the seesaw. This is a free body diagram which includes the lever arms. FN = 1500 N Next, we define the axis of rotation (circle with a dot in the middle) and the lever arms. We place all the forces at their proper location. x2 = 2.0 m x1 = 2.0 m c.g. . F1 = 400 N FL = 700 N F2 = 400 N Στ = 0 or ΣτCCW = ΣτCW F1x1 = F2x2 FN = 1500 N (400 N)(2.0 m) = (400 N)(2.0 m) FL and FN both act through the axis of rotation, so their lever arm is zero, making their torque 0. The torques balance so the seesaw can be in static equilibrium. 800 Nm = 800 Nm x2 = 2.0 m x1 = 2.0 m c.g. . F1 = 400 N FL = 700 N F2 = 400 N It is important that the seesaw be level, so that the force applied by each of the girls is acting downward and is perpendicular to the lever arm. If the Force and the Lever Arm are not perpendicular, then the equation for the torque becomes complex. It is better that we avoid that situation. So, what do you do to balance the seesaw if the two people are not the same weight (mass)? One 400 N girl sits on one end of a seesaw that is centered on the fulcrum, is 4.0 m long, and weighs 700 N. Where must her 650 N brother sit in order for the seesaw to be in static equilibrium? ? c.g. Option #1, move the heavier person closer to the fulcrum. FL and FN both act through the axis of rotation, so their lever arms are zero. Στ = 0 or ΣτCCW = ΣτCW FGxG = FBxB FN = 1750 N xB = (800 Nm)/(650 N) (400 N)(2.0 m) = (650 N)xB xB = 1.23 m 800 = 650x B xG = 2.0 m xB = ?? m c.g. . FG = 400 N FL = 700 N FB = 650 N Option #2, move the center of gravity of the seesaw so that more of the seesaw is on the side of the lighter person, One 400 N girl sits on one end of a 4.0 m long seesaw weighing 700 N That has moved the center of gravity of the lever 0.2 meters towards her. Where must her 650 N brother sit in order for the seesaw to be in static equilibrium? ? c.g. Now the weight of the seesaw creates a torque helping the girl. FN acts through the axis of rotation, so its lever arm is zero. Στ = 0 or ΣτCCW = ΣτCW FGxG + FLxL = FBxB FN = 1750 N (400)(2.2) + (700)(0.2) = (650 N)xB xB = (1020 Nm)/(650 N) 880 + 140 = 650xB xB = 1.57 m xG = 2.2 m xB = ?? m c.g. . xL = 0.2 m FG = 400 N FL = 700 N FB = 650 N For the Middle School (Division B) Competition, you will need to build a simple first class lever system. The lever may not be longer than 1.00 meter. Simple Machines. (Simple case.) Given small mass placed on one side. Given unknown large mass on the other. Unless the values are too extreme, you may be able to move the large mass close enough to the fulcrum. c.g. If this is the setup, you don’t have to worry about the weight of the lever. Simple Machines. (Simple case.) c.g. Big Mass Small Mass FN xS xB c.g. . FS FL FB Simple Machines. (Simple case.) In this case the torque equation is: τCCW = τCW (FS)(xS) = (FB)(xB) and you can solve for any value. FN xS xB c.g. . FS FL FB So far we have been dealing with the force applied by the hanging mass. This force is known as the weight of the object or the force of gravity (Fg) acting on the object. The force of gravity acting on an object is the product of the mass of the object multiplied by the gravity constant on the planet Earth (9.8 N/kg). Fg = mg = m(9.8 N/kg) so, mass, m = Fg/(9.8 N/kg) This gets quite confusing because weight is measured in newtons and mass is measured in grams or kilograms (kg). You may have been told to weigh something but you actually measured its mass in grams. Simple Machines. (Simple case.) Knowing that Fg = mg (FS)(xS) = (FB)(xB) This equation can be written: (msg)(xS) = (mBg)(xB) Dividing by g we get: (msg)(xS)/g = (mBg)(xB)/g FN xS (ms)(xS) = (mB)(xB) xB c.g. . FS = mSg FL F B = m Bg Simple Machines. (Simple case.) We can now solve for a mass using this equation and modify our torque diagram as shown: (ms)(xS) = (mB)(xB) FN xS xB c.g. . mS FL mB Simple Machines. (Simple case.) Suppose that you were given a small mass of 125 grams and an unknown large mass. You set up your lever so it balances as shown: FN xB = 10.0 cm xS = 47.6 cm c.g. . mS = 125 g FL mB = ?? Simple Machines. (Simple case.) (ms)(xS) = (mB)(xB) (125 g)(47.6 cm) = (mB)(10.0 cm) mB = (125 g)(47.6 cm)/(10.0 cm) = 595 grams FN xB = 10.0 cm xS = 47.6 cm c.g. . mS = 125 g FL mB = ?? Simple Machines (More realistic case) If the difference between the unknown mass and the known mass is large, move the fulcrum near one end of the lever. Place the unknown large mass on the short side. Then place the small mass on the long side. c.g. Now the lever helps balance the large weight. FN acts through the axis of rotation, so its lever arm is zero. Στ = 0 or ΣτCCW = ΣτCW FSxS + FLxL = FBxB FN xB = cm xS = cm c.g. . xL = cm FS = N FL = N FB = ?? N FN acts through the axis of rotation, so its lever arm is zero. Στ = 0 or ΣτCCW = ΣτCW FSxS + FLxL = FBxB FN (mSg)xS + (mLg)xL = (mBg)xB If you divide through by “g” you get: xS = cm xB = cm c.g. . mSxS + mLxL = mBxB FS = mSg FL = mLg xL = cm FB = mBg Sample: Suppose that you have set up your 1.00 meter long lever of mass 83.4 grams so that the center of gravity is FN 20.0 cm from the fulcrum. You have also determined that the big unknown mass will be placed 10.0 cm from the fulcrum. xB = 10.0 cm xS = cm c.g. . xL = 20.0 cm FS = mSg FL = mLg FB = mBg You are given a known mass of 76.2 grams and a big mass. Putting the masses on the lever, you find that it balances when the little mass is 62.8 centimeters from the fulcrum. FN xB = 10.0 cm xS = 62.8 cm c.g. . xL = 20.0 cm mS = 76.2 g mL = 83.4 g mB = ?? mSxS + mLxL = mBxB (76.2 g)(62.8 cm) + (83.4 g)(20.0 cm) = mB(10.0 cm) FN mB = [(76.2 g)(62.8 cm) + (83.4 g)(20.0 cm)]/(10.0 cm) mB = 645 grams xS = 62.8 cm xB = 10.0 cm c.g. . xL = 20.0 cm mS = 76.2 g mL = 83.4 g mB = ?? As long as the levers are horizontal and in static equilibrium, you can use the equations with mass instead of force or weight. Your Physics teacher probably will not be too happy, but the equation is mathematically correct for the situation. c.g. c.g. B First Class Lever Second Class Lever For the High School Competition you will need to build a compound lever system made up of a first class lever connected to a second class lever. Each lever may not be longer than 50.0 cm. c.g. First part of lever system mSxS + mL1xL1 = ExE1 xE1 = cm xS = cm c.g. . xL1 = cm mS = g mL1 = g E = ?? g Second part of lever system c.g. ExE2 = mL2xL2 + mBxB B xL2 = cm xE2 = cm c.g. . E= g mL2 = g xB = cm mB = ?? g Sample: c.g. xE1 = 5.0 cm xE2 = 35.0 cm c.g. xB = 10.0 cm B Suppose that you built your lever system so that the fulcrum in the class 1 lever was 5.0 cm from the string connecting the levers. Also suppose that in the second class lever the string connecting the levers is 35.0 cm from its fulcrum and you set up the lever so that the unknown big mass is 10.0 cm from the fulcrum. Sample xE1 = 5.0 cm c.g. xE2 = 35.0 cm c.g. xS = 32.7 cm xB = 10.0 cm B You are given a small mass of 86.0 grams and an unknown large mass that you place at the 10.0 cm mark. You slide the small mass along the class 1 lever and manage to get the levers to balance when the small mass is 32.7 cm from its fulcrum. xE1 = 5.0 cm c.g. xS = 32.7 cm First part of lever system You also measured the mass of the lever as 27.4 g and arranged the lever so that its center of gravity is 16.0 cm from the fulcrum. xE1 = 5.0 cm xS = 32.7 cm c.g. . xL1 = 16.0 cm mS = 86.0 g mL1 = 27.4 g mSxS + mL1xL1 = ExE1 E = ?? g First part of lever system mSxS + mL1xL1 = ExE1 (86.0 g)(32.7 cm) + (27.4 g)(16.0 cm) = E(5.0 cm) E = [(86.0 g)(32.7 cm) + (27.4 g)(16.0 cm)]/(5.0 cm) E = 650 g xE1 = 5.0 cm x = 32.7 cm S c.g. . xL1 = 16.0 cm mS = 86.0 g mL1 = 27.4 g E = ?? g xE2 = 35.0 cm Suppose the mass of the lever is 31.4 g and its center of gravity is 18.0 cm from the fulcrum. c.g. xB = 10.0 cm Second part of lever system B xE2 = 35.0 cm xL2 = 18.0 cm c.g. . xB = 10.0 cm E = 650 g mL2 = 31.4 g ExE2 = mL2xL2 + mBxB mB = ?? g Second part of lever system ExE2 = mL2xL2 + mBxB (650 g)(35.0 cm) = (31.4 g)(18.0 cm) + mB(10.0 cm) mB = [(650 g)(35.0 cm) - (31.4 g)(18.0 cm)]/(10.0 cm) mB = 2218 g = 2.218 kg xL2 = 18.0 cm xE2 = 35.0 cm c.g. . xB = 10.0 cm E = 650 g mL2 = 31.4 g mB = ?? g Things to note: • You must build your own lever system. • You may want to have two set places to have your fulcrum depending on the given masses. • You may want to have the unknown mass at a predetermined spot and thus notching the lever at that point. • Make sure that you know the mass of your lever and have marked the location of its center of gravity. Fixed Pulley A fixed pulley is basically a First Class Lever that can rotate. The mechanical advantage is 1. All a fixed pulley does is change the direction of the force. Effort Load Fulcrum Effort Load Movable Pulley A movable pulley is basically a Second Class Lever that can rotate. The mechanical advantage is 2. Effort Effort Load Load Fulcrum Block & Tackle A Block and Tackle is a combination of a movable pulley connected to a fixed pulley. In this case the mechanical advantage of the movable pulley is 2 and the MA of the fixed pulley is 1. Combined the mechanical advantage is 2. In order to calculate the Ideal Mechanical Advantage, IMA, of a Block and Tackle, you count the number of supporting ropes. Effort Load Block & Tackle A Block and Tackle is a combination of a movable pulley connected to a fixed pulley. In this case the mechanical advantage of the movable pulley is 2 and the MA of the fixed pulley is 1. Combined the mechanical advantage is 2. Effort Load Work is done to lift the Load, but you also must lift the movable pulley with it. The AMA will be less, not only because of friction in the system but because the weight of the movable pulley also has to be lifted by the Effort. dE Wheel and Axle dL A Wheel and Axle is two different diameter cylinders on the same shaft. This also is a first class lever that can rotate. The Ideal Mechanical Advantage is the ratio of the diameters. IMA = dE/dL. Load Effort Effort Load Fulcrum All these are examples of a wheel and Axle. Inclined Plane x h An inclined plane is basically a ramp that is stationary. A load is pushed up the ramp instead of being lifted straight up. The Ideal Mechanical Advantage of a ramp is the ratio of the length of the ramp (x) to the height of the ramp (h). IMA = x/h Wedge A wedge is like an inclined plane, but instead of being stationary the wedge is driven into something or between things. All these are examples of wedges. LEVERS. Class 1 Levers. Class 1 Levers. One of these can also be used as a class 2 Lever. Which one? How? COMPOUND MACHINES. This is actually a compound machine. The teeth are wedges that cut into the wood. The whole blade is a wheel and axle. These too are compound machines. The teeth are wedges that cut the wires. The handles make a class 1 lever.