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Center of Gravity and Rotational Mechanics - Chapter 8 Center of Gravity: point located at the object's ___________ position of weight distribution. When solving problems ____ of the weight of the object is considered to be acting at the center of gravity (see figure 10.7 pg 139, fig. 8.25 pg 140). Center of mass is a point where the average mass of the object is located (these are same unless the object is tall enough for gravity to act unequally in different parts of the object). 1. Throw an object and its center of mass will follow a smooth _____________ curve (see figure 10.1 and 10.4 pg. 136-137, fig. 8.21 and 8.23 pg 139). 2. Center of gravity lies in different locations for different objects (some have a center of gravity where there is no ____________). See figures 10.2,3,5,9. Fig. 8.22, 26, 28 3. Locating the center of gravity can be found through the use of a ________ bob. (fig. 10.8 pg. 139, fig 8.26 pg 140) Toppling: When the center of gravity of an object extends _____________ the base of support of the object, then the object will topple (see figure 10.10-12 pg. 140, figs. 8.29-32 pg 142). Stability: The stability of an object depends upon the _____________ of the center of gravity of the object relative to the object. There are 3 ways of looking at stability: 1. Unstable equilibrium - When any movement __________ the center of gravity. 2. Stable equilibrium - When the center of gravity of the object must be _________ (this will require work to increase the object's potential energy) to reach an unstable state. 3. Neutral equilibrium - When any displacement will not _________ the location of the center of gravity. (see figures 10.14-10.16 pg. 142-3, fig 8.36 pg 144). Objects are particularly stable when the center of gravity lies __________ the base of support (see figures 10.17-10.19). Systems tend to seek the ___________ possible position for the center of gravity. That is why a ball will rise from the container of dry beans when shaken (or crunch berries will rise to the top when you shake the box). See figure 10.20-21. Torque: pgs 137-8 Product of force applied ___ to an axis of rotation and the distance which the force is applied from the axis of rotation (called the _________ arm). = F. d Torque = perpendicular force . lever arm distance See figures 11.1-4 pg. 150-151. Question at top of 152 Balanced Torques: When torques acting clockwise and counterclockwise are _________. See fig. 11.5 pg. 152 and computational example pg. 153 1. Torques produce ________________ (much like forces produce accelerations). 2. When a football is kicked in line with the center of gravity (which is the axis of rotation), the object will not ____________ end to end (no rotation due to no lever arm distance). If struck ___________ the center of gravity, the object will topple (toppling is a rotation and rotations are produced by torques). See figure 11.7-8 pg. 154. Homework: Day #1: Review Questions pg. 154 #13-22, One Step Calculations pg 156 #1,2 Day #2: Day #2: Exercise pg. 157-8 #18, 20, 21, 28-30, 34, 36 Problems pg. 159-60 #3, 5, 6 and backside of sheet Worksheet on Torque and Center of Gravity: 1. Complete the data for the three seesaws in equilibrium. 2. The broom balances at its center of gravity. If you were to cut the broom at this point and weigh each part of the broom, how would the weights of the parts compare? 3. Balance the torque system and determine the weight in Newtons that the scale would read. The hanging masses are measured in kilograms. Neglect the mass of the hangers. Each level will have a hanger connected to it from below and will be tied to a new point on the hanger above it. Example: B to E F to __ __ to __ __ to __ __ to Scale Scale Reads ______ N Torque Problems Torque – measurement of the tendency of a force to produce a rotation about an axis. Torque = perpendicular force • lever arm τ = F┴ • d Balancing torques about a pivot point (fulcrum) τ (clockwise) = τ (counterclockwise) Center of gravity- a point on any object that acts as the place at which all the weight is concentrated Example 1: How much torque does Jed produce when he tightens a bolt by exerting 12 N of force on a wrench at a distance of 0.40 m from the fulcrum? Example 2: Mary Ann and Ginger are seesawing on the island on a seesaw made by the Professor. Mary Ann weighs 400. N and sits 2.00 m from the fulcrum of the seesaw. Where would 450. N Ginger need to sit to balance the seesaw? Example 3: The center of gravity of a 3.0 m wood plank is at its center. If the wood plank is set across a saw horse to give a pivot point of 2.0 m for one end, how much weight would need to be set on 0.25 m in from the other end of the plank in order to balance it? (mass of plank is 10.0 kg) Problems: 1. A water faucet is turned on when a force of 2.0 N is exerted on the handle, at a distance of 0.060 m from the pivot point. How much torque must be produced to turn the handle? 2. Nancy, whose mass is 60.0 kg, is working at a construction site and she sits down for a bite to eat at noon. If Nancy sits on the very end of a 3.00-m-long plank pivoted in the middle on a saw horse, how much torque must her co-worker provide on the other end of the plank in order to keep Nancy from falling on the ground? 3. Barry carries his tray of food to his favorite cafeteria’s table for lunch. The 0.50 m-long tray has a mass of 0.20 kg and holds a 0.40-kg plate of food 0.20 m from the right edge of the tray. Barry holds that tray by the left edge with one hand, using his thumb as the fulcrum, and pushes up 0.10 m from the fulcrum with his finger tips. a) How much upward force must his finger tips exert to keep the tray level? b) How might Barry make the tray easier to carry if he still chooses to use only one hand? 4. Tracey is building a mobile to hang over her baby’s crib. She hangs a 0.020-kg toy sailboat 0.010 m from the left end and a 0.015-kg toy truck 0.20 m from the right end of a bar 0.50 m long. If the lever arm itself has negligible mass, where must the support string be placed so that the arm balances? 5. Orin and Ann, two paramedics, rush a 60.0-kg man from the scene of an accident to a waiting ambulance, carrying the man on a uniform 3.00-kg stretcher held by the ends. The stretcher is 2.60 m long and the man’s center of mass is 1.00 m from Ann. How much force must Orin and Ann each exert to keep the man horizontal? Torque Problems Day 2 1. Most door knobs are placed on the side of the door opposite the hinges instead of in the center of the door. a. Why is this so? b. If a torque of 1.2 Nm is required to open a door, how much force must be exerted on a door knob 0.76 m from the hinges compared to a door knob in the middle of the door, 0.38 m from the hinges? 2. Cilla is working out in the gym with a 2.00 kg mass that she holds in one hand and gradually lifts up and down. a. Will Cilla find it easier to lift the mass if she pivots her arm at the shoulder or at the elbow? b. If Cilla’s arm is 0.60 m long from her shoulder to her palm and 0.28 m long from her elbow to her palm, how much torque must she produce in each case to lift the weight? 3. What force must be applied to balance the lever system shown? Mass of board = 20.0 kg 20.0 kg 0.50m 1.0 m 2.0 m 4. A 3.0 kg board is 5.0 m long and it rests on a fulcrum 1.5 m from the left end. A load of 10.0 kg is placed 1.0 m from the right end. If a second load is placed 0.50 m from the left end and balances the board, what is second load’s weight? ? 10.0 kg 1.50m 1.0 m 0.50 m 5. Balance the Mobile: H I J K 6. Find the masses. 30.0 cm 24 kg M3 = ? 15.0 cm D E F G 40.0 cm 9 kg 25.0 cm 20.0 cm 15.0 cm M2 = ? A 1 kg B C 2 kg 0.10 kg M1 = ? Lever Lab: Name ____________________________________________ Show the calculation of all lever arm distances and show all factor-label conversion!!!!!!!!! 1st Class Lever: Fulcrum at a point between the forces. 1. Place the 50.0 cm mark of the meter stick on the fulcrum and place a 200. g mass at the 70.0 cm mark of the stick. Place another 200. g mass on the other end of the stick to balance the stick. What is the mark to the nearest cm 0.1 cm? ___________ cm Calculate the torque on each side of the fulcrum (Remember torque is Force . distance (Nm). It is not mass . distance. Weight is a force wt = mg and N= kg . m/s2) a. Determine torque on one side of fulcrum below: b. Determine torque on other side of fulcrum below: 2. Place the 200. g mass at the 70.0 cm mark, then place the 100. g mass at a mark which will balance the stick. What is the mark to the nearest 0.1 cm? ___________ cm Calculate the torque on each side of the fulcrum (show calculation below) a. Determine torque on one side of fulcrum below: b. Determine torque on other side of fulcrum below: 3. Place the 200. g mass at the 60.0 cm mark, then place the unknown weight at a mark which will balance the stick. What is the mark to the nearest 0.1 cm? Calculate the mass of the unknown (show calculation below). ___________ cm 4. Measure the weight of the meter stick using a spring scale to the nearest 0.10 N (this will be used as the true value to calculate all % errors) Place the meter stick at the 75.0 cm mark of the fulcrum. Then place the 200. g mass at a position to balance the stick. What is the mark to the nearest 0.1 cm? Calculate the weight of the meter stick (show calculation below) ____________ N ___________ cm Determine % error 5. Place the meter stick on the table and lift the meter stick with the spring scale (holding it perfectly vertical)using the 0.0 cm mark of the meter stick as a fulcrum. What is the reading of the spring scale to the nearest 0.10 N? _____________ N Determine the upward torque of your pull (show calculation below) Using this upward torque and using the center of gravity of the meter stick to be at 50.0 cm, calculate the weight of the meter stick (show calculation below) Determine % error 2nd Class Lever: 6. Place the 200. g mass at the 90.0 cm mark, place the spring scale into the end of the meter stick and lift the stick using the 0.0 cm mark as the fulcrum. What is the force on the spring scale to the nearest 0.10 N? ______________ N Using this data, determine the upward torque from your pull (show calculation below) Using this date, determine the downward torque from the 200. g mass and the meter stick itself (use the weight of the stick measured in step 4 of 1st class levers) How do the 2 torques compare? 7. Place the 200. g mass at the 50.0 cm mark, place the spring scale into the end of the meter stick and lift the stick using the 0.0 cm mark as a fulcrum. What is the force on the spring scale to the nearest 0.10 N? Using this data, determine the weight of the meter stick (show below) _______________ N Determine % error Rotational Mechanics Review Sheet Center of Gravity 1. We can consider _______ of the weight of the object to be acting at its _______________ when doing problems. 2. When an object is thrown, its center of mass follows a ______________. 3. The center of gravity of a boomerang lies _________ its’ geometric bounds. Toppling 4. When the center of gravity extends beyond the base of support for a stationary object, it will __________. Stability 5. When any movement lowers the center of gravity, it is considered to be ___________. 6. When any movement raises the center of gravity, it is considered to be ____________. 7. When any movement will not change the location of the center of gravity, it is considered to be ____________. 8. Systems tend to seek the ___________ possible position for the center of gravity. (give an example) Torque 9. A change in torque can be produced by a change in ________ or _________. Balanced Torques 10. When torques acting clockwise and counterclockwise are _______. 11. When a football is kicked outside the center of gravity the football will _____________. Problems: 1. Mike is trying to loosen a nut with a wrench by applying a 35 N force that is perpendicular to the level arm. If it takes 25 N.m of torque in order to loosen the nut, how long must the lever arm be from the fulcrum. 2. If a torque of 1.2 Nm is required to open a door, how much force must be exerted on a door knob 0.76 m from the hinges compared to a door knob in the middle of the door, 0.38 m from the hinges? What force must be applied to balance the lever system shown? Mass of board = 35.0 kg 15.0 kg 0.40m 4. 1.5 m Balance the Mobile and tell what the spring scale would read. H I J K 3. 24 kg D E F G 6 kg A B C 4 kg 5. 2 kg A 5.0 kg board is 3.0 m long and it rests on a fulcrum 1.0 m from the left end. A load of 7.0 kg is placed 1.0 m from the right end. If a second load is placed 0.35 m from the left end and balances the board, what is second load’s weight? ? 1.0m 0.35 m 7.0 kg 1.0 m Answers: 1. all, center of gravity 2. smooth, parabolic path 3. outside 4. topple 5. unstable 6. stable 7. neutral 8. lowest 9. force perpendicular to lever arm; lever arm distance 10. equal 11. topple end to end 1. d = τ / F d = 25 Nm / 35 N d = 0.71 m 2. F = τ / d F = 1.2 Nm / 0.76 m F = 1.6 N F=τ/d F = 1.2 Nm / 0.38 m F = 3.2 N 3. τ ↑ = τ ↓ (Fd)puah = (mgd) mass + (mgd)board F push= ((mgd) + (mgd)board) / dpush F push= ((15.0 kg)(9.80 m/s2)(0.40 m) + (35.0 kg)(9.80 m/s2)(0.75 m)) / 1.5 m F push = (58.80 Nm + 257.25 Nm ) / 1.5 m F push = 316.05 Nm / 1.5 m F push= 210 N 4. B to F or another way to balance G to K I to scale Scale reading: wt = mg wt = (36 kg (9.800 m/s2)) wt = 350 N B to D F to K I to scale 5. τ = τ (F.d)unknown = (F.d)board + (F.d)mass F . = ((mgd)board + (mgd)mass) / dunknown F = ((5.0 kg)(9.80 m/s2)(0.5 m) + (7.0 kg) (9.80 m/s2) (1.0 m)) / 0.65 m F = (24.5 Nm + 68.6 Nm) / 0.65 m F = 93.1 Nm / 0.65 m F = 143 N F = 100 N