Push on an object that is free to move, and you

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1 ROTATIONAL EQUILIBRIUM Objectives Describe how to make an object turn or rotate. (11.1) Explain what happens when balanced torques act on an object. (11.2) Describe how to find an object s center of mass. (11.3) Describe how the center of gravity of an everyday object is related to its center of mass. (11.4) Describe how to predict whether an object will topple. (11.5) Explain why the center of gravity of a person is not located in a fixed place. (11.6) Describe what happens to the center of gravity of an object when the object is toppled. (11.7) discover! 11 ROTATIONAL EQUILIBRIUM THE BIG IDEA An object remains in rotational equilibrium if its center of mass is above the area of support. Push on an object that is free to move, and you set it in motion. Some objects will move without rotating, some will rotate without moving, and others will do both. For example, a kicked football often tumbles end over end. What determines whether an object will rotate when a force acts on it? Why doesn t the Leaning Tower of Pisa rotate and topple over? This chapter is about the factors that affect rotational equilibrium. We will see that these factors explain most of the techniques used by gymnasts, ice skaters, skateboarders, and divers. MATERIALS salt, small beaker EXPECTED OUTCOME Students will find that they can balance a beaker on salt particles. ANALYZE AND CONCLUDE 1. The salt particles, which offer a torque that counters that produced by gravity 2. Only a few grains 3. An object won t topple if there are no unbalanced forces or unbalanced torques acting on it. discover! How Far Can Objects Be Tipped Before They Topple Over? 1. Pour a teaspoon of salt onto a flat surface. 2. Place the base of a small beaker or flatbottomed drinking glass on the salt. 3. While tilting the beaker to the side, gently work the base of the beaker into the salt. 4. With a little finesse, the beaker will remain leaning when you remove your hand. 5. Blow away as much salt as you can without disturbing the beaker. Analyze and Conclude 1. Observing What prevents the beaker from toppling over? 2. Predicting What do you think would be the least amount of salt needed to support the beaker? 3. Making Generalizations How can you ensure an object won t topple over?

2 11.1 Torque Every time you open a door, turn on a water faucet, or tighten a nut with a wrench, you exert a turning force. These everyday movements are shown in Figure Torque is produced by this turning force and tends to produce rotational acceleration. Torque is different from force. If you want to make an object move, apply a force. Unbalanced forces make things accelerate. To make an object turn or rotate, apply a torque. Torques produce rotation. In Chapter 2 we learned that systems are in mechanical equilibrium when F 0. The other condition for mechanical equilibrium is the rotational part: torques Torque Key Terms torque, lever arm Common Misconception Torque and force are the same concept. FACT Forces cause acceleration, and torques cause rotation. Demonstration Place an L-shaped object on the table and show how it topples in different positions. FIGURE 11.1 A torque produces rotation. A torque is produced when a force is applied with leverage. You use leverage when you use a claw hammer to pull a nail from a piece of wood. The longer the handle of the hammer, the greater the leverage and the easier the task. The longer handle of a crowbar provides even more leverage. You use leverage when you use a screwdriver or a table knife to open the lid of a paint can. A torque is used when opening a door. A doorknob is placed far away from the turning axis at its hinges to provide more leverage when you push or pull on the doorknob. The direction of your applied force is important. In opening a door, you d never push or pull the doorknob sideways to make the door turn. As shown in Figure 11.2, you push perpendicular to the plane of the door. Experience has taught you that a perpendicular push or pull gives more rotation for less effort. think! If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown? Answer: 11.1 FIGURE 11.2 When a perpendicular force is applied, the lever arm is the distance between the doorknob and the edge with the hinges. Teaching Tip Demonstrate torques at work such as prying a lid off a can with a screwdriver, turning a nut with a wrench, or even opening (rotating) a door. Explain that a steering wheel is simply a modified wrench, and explain why trucks and heavy vehicles without power steering use large-diameter steering wheels. In all these cases, there are two important considerations: the application of a force and leverage. Teaching Tip Pass around a meterstick with a weight suspended at one end. Have students hold the stick horizontally and note the different torques when the weight s distance from the center of the stick is varied. CHAPTER 11 ROTATIONAL EQUILIBRIUM

3 We say torque 5 Fd, and in the previous chapter we said work 5 Fd. The distance for torque is not the same as the distance used for work. In work, the distance d is the distance the force moves (parallel to the force). In torque, the distance d refers to the leverage distance (perpendicular to the force). If you have used both short- and long-handled wrenches, you also know that less effort and more leverage result with a long handle. When the force is perpendicular, the distance from the turning axis to the point of contact is called the lever arm. If the force is not at a right angle to the lever arm, then only the perpendicular component of the force, F, will contribute to the torque. Torque is defined as 11.1 torque force lever arm CONCEPT a torque. CHECK To make an object turn or rotate, apply Teaching Resources Reading and Study Workbook Concept-Development Practice Book 11-1 Problem-Solving Exercises in Physics 7-2 Laboratory Manual 36 Transparency 16 PresentationEXPRESS Interactive Textbook Next-Time Question 11-1 FIGURE 11.3 Although the magnitudes of the applied forces are the same in each case, the torques are different. So the same torque can be produced by a large force with a short lever arm, or a small force with a long lever arm. Similarly, as shown in Figure 11.3, the same force can produce different amounts of torque. Greater torques are produced when both the force and lever arm are large. CONCEPT CHECK How do you make an object turn or rotate? discover! MATERIALS spool of thread EXPECTED OUTCOME When the thread is pulled horizontally to the right, the torque is clockwise and the spool rolls to the right. Pulled straight up the torque is counterclockwise and the spool rolls to the left. THINK Yes; at the angle where the line of action of the string intersects the point at which the spool touches the table, there is no lever arm and therefore no torque. 190 Can You Pull a String Without Producing Torque? 1. Place a spool of string or thread on a table. For best results, use a spool with rims noticeably wider than its axle. 2. Pull gently on the string or thread so that the spool rolls without skidding and its gain in rotational speed is directly proportional to the torque. 3. Predict the effect of pulling the string both ways with the string on the top and with the string on the bottom. 4. Think Is there an angle at which the string can be pulled that will produce no torque? 190 discover!

4 11.2 Balanced Torques Torques are intuitively familiar to youngsters playing on a seesaw. Children can balance a seesaw even when their weights are not equal. Weight alone does not produce a change in rotation torque does. Children soon learn that the distance they sit from the pivot point is as important as their weight. In Figure 11.4, the heavier boy sits a shorter distance from the fulcrum (turning axis) while the lighter girl sits farther away. Balance is achieved if the torque that tends to produce clockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the girl. When balanced torques act on an object, there is no change in rotation. do the math! FIGURE 11.4 A pair of torques can balance each other Balanced Torques Teaching Tip Extend rotation to seesaws. Show this with a sketch of equal-weight players (similar to Figure 11.4). Show how the force 3 distance on the left side of the fulcrum that tends to make the seesaw rotate counterclockwise is equal to the force 3 distance on the right side. Discuss the case of the twice-as-heavy boy in Figure Ask If the boy in Figure 11.4 weighed 600 N, how far would he have to sit from the fulcrum for equilibrium? 1 m Demonstrations What is the weight of the block hung at the 10-cm mark? The meterstick is supported at the center, and a 20-N block is hung at the 80-cm mark. The block hung at the 10-cm mark just balances the system. You can compute the unknown weight by applying the principle of balanced torques. The block of unknown weight tends to rotate the system of blocks and stick counterclockwise (ccw), and the 20-N block tends to rotate the system clockwise (cw). The system is in balance when the two torques are equal: counterclockwise torque clockwise torque (F d) ccw (F d) cw Rearrange the equation to solve for the unknown weight: F ccw? (F ) cw (d) cw (d) ccw The lever arm for the unknown weight is 40 cm, because the distance between the 10-cm mark and the pivot point at the 50-cm mark is 40 cm. Similarly, the lever arm for the 20-N block is 30 cm because its distance from the pivot point is 30 cm. Substituting these values into the equation, we determine the unknown weight: F ccw (20 N) (30 cm) (40 cm) 15 N The unknown weight is thus 15 N. This makes sense. You can tell that the weight is less than 20 N because its lever arm is greater than that of the block of known weight. In fact, the unknown weight s lever arm is (40 cm) (30 cm) or 4 3 that of the first block, so its weight is 3 4 as much. Anytime you use physics to compute something, consider whether or not your answer makes sense. Computation without comprehension is not conceptual physics! Demonstrate balanced torques with a meterstick balance. Adjust the masses hung and the positions of the fulcrum. Relate this to a seesaw. Make a candle seesaw. Trim a candle so that the wick is exposed at both ends. Balance the candle with a needle through the center. Rest the ends of the needle on a pair of drinking glasses. Light both ends of the candle. As the wax drips, the CG shifts, causing the candle to oscillate. CHAPTER 11 ROTATIONAL EQUILIBRIUM

5 CONCEPT When balanced CHECK torques act on an object, there is no change in rotation. Teaching Resources Reading and Study Workbook Laboratory Manual 38, 39, 40 PresentationEXPRESS Interactive Textbook 11.3 Center of Mass Key Term center of mass Teaching Tip Define center of mass as the average position of mass, and center of gravity as the average position of weight. For our purposes, they describe the same point. Show that the center of mass of a book is at the geometrical center, which is easily found by the intersection of diagonal lines from opposite corners. Teaching Tip Point out that though a baseball bat is an irregularly shaped object, it has an axis of symmetry that runs along the bat s length. A ball, on the other hand, has axes of symmetry in all spatial directions. FIGURE 11.5 This scale relies on balanced torques. Scale balances that work with sliding weights, such as the one shown in Figure 11.5, are based on balanced torques, not balanced masses. The sliding weights are adjusted until the counterclockwise torque just balances the clockwise torque. Then the arm remains horizontal. We say the scale is in rotational equilibrium. CONCEPT CHECK What happens when balanced torques act on an object? 11.3 Center of Mass Throw a baseball into the air, and it follows a smooth parabolic path. Throw a baseball bat into the air and its path is not smooth. The bat seems to wobble all over the place. But it wobbles about a special point. As shown in Figure 11.6, this point stays on a parabolic path, even though the rest of the bat does not. The motion of the bat is the sum of two motions: (1) a spin around this point and (2) a movement through the air as if all the mass were concentrated at this point. This point, called the center of mass, is where all the mass of an object can be considered to be concentrated. Location of the Center of Mass The center of mass of an object is the point located at the object s average position of mass. The center of mass of various objects in Figure 11.7 is shown by a dot. For a symmetrical object, such as a baseball, this point is at the geometric center of the object. But an irregularly shaped object, such as a baseball bat, has more mass at one end than the other end, so the center of mass is toward the heavier end. The center of mass of a piece of tile cut into the shape of a triangle is located on the line passing through the center and the apex, one-third of the way up from the base. A solid cone s center of mass is one-fourth of the way up from its base. FIGURE 11.6 The centers of mass of the baseball and of the spinning baseball bat each follow parabolic paths

6 Objects not made of the same material throughout (that is, objects of varying density) may have the center of mass quite far from the geometric center. Consider a hollow ball half filled with lead. The center of mass would not be at the geometric center; rather, it would be located somewhere within the lead part. The ball will always roll to a stop with its center of mass as low as possible. Make the ball the body of a lightweight toy clown, and whenever it is pushed over, it will come back right-side up as illustrated in Figure FIGURE 11.7 The center of mass for each object is shown by the colored dot. Demonstrations Show a piece of irregularshaped plywood (somewhat larger than this book) that has about five corners. Fasten three short pieces of string at different places along its edge. Ask how the center of mass of this shape can be found. Suspend the plywood from one of the strings and draw a vertical chalk line beneath the point of suspension. Do this with a second string and the intersection is the center of mass. Double-check by suspending it from a third string. FIGURE 11.8 The center of mass of the toy is below its geometric center. Motion About the Center of Mass The multiple-flash photograph in Figure 11.9 shows the top view of a wrench sliding across a smooth horizontal surface. Notice that its center of mass, marked by the white dot, follows a straight-line path. Other parts of the wrench rotate about this point as the wrench moves across the surface. The motion of the wrench is a combination of straight-line motion of its center of mass and rotation around its center of mass. FIGURE 11.9 The center of mass of the rotating wrench follows a straight-line path. Attempt to balance an L- shaped piece of plywood with its small end on the table and watch it topple. That s because it wasn t supported at its average position of mass its center of mass. Illustrate center of mass with a baseball bat and other objects. Toss a small ball from one hand to the other and call attention to the smooth parabola it traces. Then toss an L-shaped piece of wood or other material. State that in doing so it doesn t seem to follow a smooth parabola that it wobbles. In fact, it wobbles about the center of mass. CHAPTER 11 ROTATIONAL EQUILIBRIUM

7 Teaching Tidbit Planets orbiting stars are detected not only by star wobble, but by the slight dimming of starlight seen when a planet crosses in front of the star. FIGURE The center of mass of the fireworks rocket and its fragments move along the same path before and after the explosion. If the wrench were instead tossed into the air, no matter how it rotated, its center of mass would follow a smooth parabola. The same is true even for an exploding projectile, such as the fireworks rocket shown in Figure The internal forces during the explosion do not change the projectile s center of mass. Interestingly enough, if air resistance is negligible, the center of mass of the dispersed fragments as they fly through the air will be at any time where the center of mass would have been if the explosion had never occurred. For: Links on center of mass Visit: Web Code: csn 1103 Applying Spin to an Object When you throw a ball and apply spin to it, or when you launch a plastic flying disk, a force must be applied to the edge of the object. This produces a torque that adds rotation to the projectile. If you wish to kick a football so that it sails through the air without tumbling, kick it in the middle, as illustrated in Figure 11.11a. If you want it to tumble end over end in its trajectory, kick it above or below the middle, as shown in Figure 11.11b. Then you apply torque as well as force to the ball. A skilled pool player similarly strikes the cue ball below its center to put backspin on the ball. CONCEPT CHECK Where is an object s center of mass located? CONCEPT The center of mass of CHECK an object is the point located at the object s average position of mass. a b Teaching Resources Reading and Study Workbook Transparency 17 PresentationEXPRESS Interactive Textbook FIGURE a. If the football is kicked in line with its center, it will move without rotating. b. If it is kicked above or below its center, it will rotate

8 11.4 Center of Gravity Center of mass is often called center of gravity, which is the average position of all the particles of weight that make up an object. For almost all objects on and near Earth, these terms are interchangeable. There can be a small difference between center of gravity and center of mass when an object is large enough for gravity to vary from one part to another. For example, the center of gravity of the Sears Tower in Chicago is about 1 millimeter below its center of mass. This is due to the lower stories being pulled a little more strongly by Earth s gravity than the upper stories. For everyday objects, the center of gravity is the same as the center of mass. Wobbling If you threw a wrench so that it rotated as it moved through the air, you d see it wobble about its center of gravity. The center of gravity itself would follow a parabolic path. Now suppose you threw a lopsided ball one with its center of gravity off-center. You d see it wobble also. The sun itself wobbles for a similar reason. As shown in Figure 11.12, the center of gravity of the solar system can lie outside the massive sun, not at the sun s geometric center. Why? Because the masses of the planets contribute to the overall mass of the solar system. As the planets orbit at their respective distances, the sun actually wobbles off-center. Astronomers look for similar wobbles in nearby stars the wobble is an indication of a star with a planetary system. Locating the Center of Gravity The center of gravity (called the CG from here on) of a uniform object (such as a meterstick) is at the midpoint, its geometric center. The CG is the balance point. Supporting that single point supports the whole object. In Figure the many small vectors represent the force of gravity along the meterstick. All of these can be combined into a resultant force that acts at the CG. The effect is as if the weight of the meterstick were concentrated at this point. That s why you can balance the meterstick with a single upward force directed at this point. FIGURE If all the planets were lined up on one side of the sun, the center of gravity of the solar system would lie outside the sun. FIGURE The weight of the entire stick behaves as if it were concentrated at its center Center of Gravity Key Term center of gravity Teaching Tip State that when the average position of weight is considered, one speaks of the center of gravity (CG). For most cases, the center of gravity and center of mass are indistinguishable, so CG will be taken to mean both. Demonstration Attach a weight to one side of a basketball. Toss it across the room and the wobble will be evident. Teaching Tip Note that Figure is not to scale. Next to the sun itself, Jupiter contributes most to the solar system s CG. Common Misconception The center of gravity of an object must be where physical mass exists. FACT The center of gravity may be located where no actual material exists. Teaching Tip Point out the CG of various objects in your classroom. In particular, try to find some objects in which the centers of gravity are located where no part of the object exists. CHAPTER 11 ROTATIONAL EQUILIBRIUM

9 Teaching Tip Have students locate the CG of the United States (contiguous 48 states) using a cutout map posted onto a cardboard backing such as that shown in Figure (They should find the CG located near Lebanon, Smith County, Kansas.) As an extension of this, have them find the CG of their home state. Demonstration Place a wind-up (spring action) stopwatch on top of an upside-down watch glass with a tiny mirror siliconed to the watch. Shine a laser beam on the mirror; watch the beam move back and forth with the rocking motion of the watch. The CG of the watch is slightly displaced with each tick! FIGURE You can use a plumb bob to find the CG for an irregularly shaped object. FIGURE11.15 There is no material at the CG of these objects. think! Where is the CG of a donut located? Answer: Can an object have more than one CG? Answer: If you suspend any object (a pendulum, for example) at a single point, the CG of the object will hang directly below (or at) the point of suspension. To locate the object s CG, construct a vertical line beneath the point of suspension. The CG lies somewhere along that line. Figure shows how a plumb line and bob can be used to construct a line that is exactly vertical. You can locate the CG by suspending the object from some other point and constructing a second vertical line. The CG is where the two lines intersect. The CG of an object may be located where no actual material exists, as illustrated in Figure The CG of a ring lies at the geometric center where no matter exists. The same holds true for a hollow sphere such as a basketball. The CG of even half a ring or half a hollow ball is still outside the physical structure. There is no material at the CG of an empty cup, bowl, or boomerang. CONCEPT CHECK How is the center of gravity of an everyday object related to its center of mass? CONCEPT For everyday objects, CHECK the center of gravity is the same as the center of mass. Teaching Resources Reading and Study Workbook PresentationEXPRESS Interactive Textbook Next-Time Question 11-2 Conceptual Physics Alive! DVDs Center of Gravity FIGURE The block topples when the CG extends beyond its support base Torque and Center of Gravity Pin a plumb line to the center of a heavy wooden block and tilt the block until it topples over as shown in Figure You can see that the block will begin to topple when the plumb line extends beyond the supporting base of the block

10 The Rule For Toppling If the center of gravity of an object is above the area of support, the object will remain upright. If the CG extends outside the area of support, an unbalanced torque exists, and the object will topple. This principle is dramatically employed in Figure The bus must not topple when the chassis is tilted 28 with the top deck fully loaded and only the driver and conductor on the lower deck. Because so much of the weight of the vehicle is in the lower part, the load of the passengers on the upper deck raises the CG only a little, so the bus can be tilted well beyond this 28 limit without toppling. The Leaning Tower of Pisa does not topple because its CG does not extend beyond its base. As shown in Figure 11.18, a vertical line below the CG falls inside the base, and so the Leaning Tower has stood for centuries. If the tower leaned far enough that the CG extended beyond the base, an unbalanced torque would topple the tower. The support base of an object does not have to be solid. The four legs of a chair bound a rectangular area that is the support base for the chair, as shown in Figure Practically speaking, supporting props could be erected to hold the Leaning Tower up if it leaned too far. Such props would create a new support base. An object will remain upright if the CG is above its base of support. FIGURE This Londoner doubledecker bus is undergoing a tilt test. FIGURE The Leaning Tower of Pisa does not topple over because its CG lies above its base. FIGURE The shaded area bounded by the bottom of the chair legs defines the support base of the chair Torque and Center of Gravity Teaching Tip Tell your students that a floating iceberg will not tip over because its CG is below the water line. If it were to tip, its CG would be raised, which requires work input. This is true for Sutro Tower in San Francisco, which easily withstands strong winds. Its base is so deeply buried in concrete that in a sense it is already tipped over. The same applies to the Space Needle in Seattle. Teaching Tip Stress that work input is needed to raise the CG of a system. Relate this idea to Figures and Teaching Tip Tell your students that 400 tons of lead ingots were stacked on the base of the Leaning Tower of Pisa. The lead is reversing the 800 years of slow tilting and is pulling the 187-ft pillar of white marble back toward the vertical. Demonstration Have a student volunteer sit on a chair in front of the class and attempt to stand up without putting his or her feet under the chair. (This cannot be done because there is no support base that lies beneath the person s CG.) CHAPTER 11 ROTATIONAL EQUILIBRIUM

11 Teaching Tip As shown in Figure 11.21, the moon rotates about its CM, while a torque due to Earth s gravity is exerted at its CG. Because of the slight distance between the CM and CG, one hemisphere of the moon always faces Earth. FIGURE Gyroscopes and computerassisted motors in the selfbalancing electric scooter make continual adjustments to keep the combined CGs of Mark, Tenny, and the vehicles above the support base. Balancing Try balancing a broom upright on the palm of your hand. The support base is quite small and relatively far beneath the CG, so it s difficult to maintain balance for very long. After some practice, you can do it if you learn to make slight movements of your hand to exactly respond to variations in balance. You learn to avoid under-responding or overresponding to the slightest variations in balance. A self-balancing electric scooter, like the one shown in Figure 11.20, does much the same. Variations in balance are quickly sensed and an internal high-speed computer regulates a motor to keep the vehicle upright. The computer regulates corrective adjustments of the wheel speed, in a way quite similar to the way your brain coordinates the adjustments you make when balancing a broom on the palm of your hand. Both feats are truly amazing. FIGURE The moon is slightly footballshaped due to Earth s gravitational pull. EXAGGERATED MOON CM CG CM LEVER ARM A TORQUE EXISTS WHEN MOON S LONG AXIS IS NOT ALIGNED WITH EARTH S GRAVITATIONAL FIELD CONCEPT If the center of CHECK gravity of an object is above the area of support, the object will remain upright. Teaching Resources Reading and Study Workbook PresentationEXPRESS Interactive Textbook Next-Time Question 11-3 The Moon s CG Center of gravity and torque explain the fact that only one side of the moon continually faces Earth. Because the side of the moon nearest Earth is gravitationally tugged toward Earth a bit more than farther parts, the moon s CG is slightly closer to Earth than its center of mass. While the moon rotates about its center of mass, Earth pulls on its CG. This produces a torque when the moon s CG is not on the line between the moon s and Earth s centers, as illustrated in Figure This torque keeps one hemisphere of the moon facing Earth, just as torque aligns a magnetic compass in a magnetic field. CONCEPT CHECK What is the rule for toppling?

12 11.6 Center of Gravity of People The center of gravity of a person is not located in a fixed place, but depends on body orientation. When you stand erect with your arms hanging at your sides, your CG is within your body. It is typically 2 to 3 cm below your navel, and midway between your front and back. The CG is slightly lower in women than in men because women tend to be proportionally larger in the pelvis and smaller in the shoulders. In children, the CG is approximately 5% higher because of their proportionally larger heads and shorter legs. Raise your arms vertically overhead. Your CG rises 5 to 8 cm. Bend your body into a U or C shape and your CG may be located outside your body altogether. This fact is nicely employed by the high jumper in Figure 11.22, who clears the bar while his CG nearly passes beneath the bar. As shown in Figure 11.23, when you stand, your CG is somewhere above your support base, the area bounded by your feet. In unstable situations, as in standing in the aisle of a bumpy-riding bus, you place your feet farther apart to increase this area. Standing on one foot greatly decreases this area. In learning to walk, a baby must learn to coor dinate and position the CG above a supporting foot. Many birds, pigeons for example, do this by jerking their heads back and forth with each step. FIGURE A high jumper executes a Fosbury flop to clear the bar while his CG nearly passes beneath the bar. FIGURE When you stand, your CG is somewhere above the area bounded by your feet Center of Gravity of People Common Misconception The center of gravity of a person is at a fixed place inside the body. FACT The CG of a person is not located in a fixed place, but depends on body orientation. Teaching Tip Explain that for most adults, the CG lies a bit below the bellybutton. In children, the CG is about 5% higher because of their proportionally larger heads and shorter legs. Explain again that the CG in men is generally higher than in women (1 2%). Teaching Tip Just as the CG of a boomerang is outside the material, a person s CG is outside the body when he or she bends over and makes a U or L shape. Whatever the body orientation, to remain stable the CG must be above (or below) a support base area. Show how this support base is enlarged when a person stands with feet wide apart (Figure 11.23). Link to BIOLOGY Tails You can bend over only so far when trying to extend your horizontal reach. How far you can extend de pends on keeping your CG within your support base. A monkey can reach proportionally much farther than you can without toppling. How? By extending its tail, thus keeping its CG above its feet. A tail gives an animal the ability to shift its CG and increase stability. The massive tails of dinosaurs tell us that they were able to extend their heads considerably beyond the support base of their feet. CHAPTER 11 ROTATIONAL EQUILIBRIUM

13 discover! MATERIALS chair EXPECTED OUTCOME Females can generally perform this feat while males generally cannot. THINK The CG in males is generally higher than in females, mainly because females tend to be proportionally smaller in the upper body and larger in the pelvic region. This CG explanation may not apply to all people. People come in a variety of sizes and proportions. discover! Do Males and Females Have the Same CG? 1. Stand exactly two footlengths away from a wall and place a chair between yourself and the wall. 2. Bend over with a straight back and let your head lean against the wall as shown. 3. Lift the chair off of the floor while your head is still leaning against the wall. 4. Now attempt to straighten up while still holding onto the chair. 5. Think Why can females generally do this while males cannot? Demonstration Ask a student to stand facing a wall with toes against the wall and simply stand unaided on tiptoes for a couple of seconds. (This cannot be done, because the CG will be farther from the wall than the support base provided by the toes. With a bit of trickery this can be done near a doorway when you hold a heavy weight from your extended arm into the doorway in front of yourself so that the CG is above the narrow support base.) FIGURE You can lean over and touch your toes without toppling only if your CG is above the area bounded by your feet. think! When you carry a heavy load such as a pail of water with one arm, why do you tend to hold your free arm out horizontally? Answer: 11.6 You can probably bend over and touch your toes without bending your knees. In doing so, you unconsciously extend the lower part of your body, as shown in Figure In this way your CG, which is now outside your body, is nevertheless above your supporting feet. If you try it while standing with your heels to a wall, you may be in for a surprise. You cannot do it! This is because you are unable to adjust your body, and your CG protrudes beyond your feet. You are off balance and torque topples you over. CONCEPT CHECK On what does the location of a person s center of gravity depend? 11.7 Stability It is nearly impossible to balance a pen upright on its point, while it is rather easy to stand it upright on its flat end, because the base of support is inadequate for the point and adequate for the flat end. But there is a second reason. Consider a solid wooden cone on a level table. As you can see in Figure 11.25a, you cannot stand it on its tip. Even if you position it so that its CG is exactly above its tip, the slightest vibration or air current will cause the cone to topple

14 a b c Change in the Location of the CG Upon Toppling When an object is toppled, the center of gravity of that object is raised, lowered, or unchanged. What happens to the CG of the cone in Figure 11.25a when it topples? The answer to this question provides the second reason for stability. A little thought will show that the CG is lowered by any movement. We say that an object balanced so that any displacement lowers its center of mass is in unstable equilibrium. A cone balances easily on its base, as shown in Figure 11.25b. To make it topple, its CG must be raised. This means the cone s potential energy must be increased, which requires work. We say an object that is balanced so that any displacement raises its center of mass is in stable equilibrium. An object that is balanced so that any small movement neither raises nor lowers its center of gravity is in neutral equilibrium. A cone lying on its side, such as the one shown in Figure 11.25c, is in neutral equilibrium. Like the cone, the pen is in unstable equilibrium when it is on its point. When the pen is on its flat end, as in Figure 11.26, it is in stable equilibrium because the CG must be raised slightly to topple it over. Consider the upright book and the book lying flat in Figure Both are in stable equilibrium. But you know the flat book is more stable. Why? Because it would take considerably more work to raise its CG to the point of toppling than to do the same for the upright book. An object with a low CG is usually more stable than an object with a relatively high CG. FIGURE a. Equilibrium is unstable when the CG is lowered with displacement. b. Equilibrium is stable when work must be done to raise the CG. c. Equilibrium is neutral when displacement neither raises nor lowers the CG. FIGURE For the pen to topple when it is on its flat end, it must rotate over one edge. During the rotation, the CG rises slightly and then falls. CONCEPT The center of gravity CHECK of a person is not located in a fixed place, but depends on body orientation. Teaching Resources Reading and Study Workbook Concept-Development Practice Book 11-2 Laboratory Manual 37 PresentationEXPRESS Interactive Textbook 11.7 Stability Key Terms unstable equilibrium, stable equilibrium, neutral equilibrium Teaching Tip Distinguish between unstable, neutral, and stable equilibriums, using the examples shown in Figure FIGURE Toppling the upright book requires only a slight raising of its CG. Toppling the flat book requires a relatively large raising of its CG. CHAPTER 11 ROTATIONAL EQUILIBRIUM

15 Teaching Tip Show that the CG of an object is either raised, not changed, or lowered when the object is tipped. Also demonstrate, if possible, the devices shown in Figures and FIGURE A pencil balanced on the edge of a hand is in unstable equilibrium. a. The CG of the pencil is lowered when it tilts. b. When the ends of the pencil are stuck into long potatoes that hang below, it is stable because its new CG rises when it is tipped. a Objects in Stable Equilibrium The horizontally balanced pencil in Figure 11.28a is in unstable equilibrium. Its CG is lowered when it tilts. But suspend a potato from each end and the pencil becomes stable, as shown in Figure 11.28b. Why? Because the CG is below the point of support, and is raised when the pencil is tilted. Some well-known balancing toys depend on this principle. Their secret is that they have been weighted so that the CG lies vertically underneath the point of support while most of the remainder of the toy is above it. See the example in Figure A toy that hangs with its CG below its point of support is in stable equilibrium because the CG rises when the toy tilts. b FIGURE The toy is in stable equilibrium because the CG rises when the toy tilts. 202 FIGURE The Seattle Space Needle can no more fall over than can a floating iceberg. 202 The CG of a building is lowered if much of the structure is below ground level. This is important for tall, narrow structures. An extreme example is the state of Washington s tallest freestanding structure, the Space Needle in Seattle, which is shown in Figure This structure is so deeply rooted that its center of mass is actually below ground level. It cannot fall over intact. Why? Because falling would not lower its CG at all. If the structure were to tilt intact onto the ground, its CG would be raised!

16 Demonstration a Lowering the CG of an Object The tendency for the CG to take the lowest position available is illustrated in Figure Place a very light object, such as a table tennis ball, at the bottom of a box of dried beans or small stones. Shake the box, and the beans or stones tend to go to the bottom and force the ball to the top. By this process the CG of the whole system takes a lower position. b FIGURE The CG of an object has a tendency to take the lowest position available. a. A table tennis ball is placed at the bottom of a container of dried beans. b. When the container is shaken from side to side, the ball is nudged to the top. FIGURE The CG of the glass of water is affected by the position of the table tennis ball. a. The CG is higher when the ball is anchored to the bottom. b. The CG is lower when the ball floats. Shake a container of dried beans with a table tennis ball at the bottom, as shown in Figure Here, the density of the beans is greater than the density of the table tennis ball, so it s like panning gold. For objects of the same density, smaller ones will fill in the open spaces between larger ones and similarly produce a greater effective density of space. The CG lowers. The above demo can be extended to the table tennis ball in a glass of water. The CG of the system is lowest when the table tennis ball floats. Push it under the surface and the CG is raised. If you do the same thing with something more dense than water, the CG is lowest when it is sunk at the bottom. (More about this in Chapter 19.) As shown in Figure 11.32, the same thing happens in water when an object rises to the surface and floats. If the object weighs less than an equal volume of water, the CG of the whole system will be lowered when the object is forced to the surface. This is because the heavier (more dense) water can then occupy the available lower space. If the object is heavier than an equal volume of water, it will be more dense than water and sink. In either case, the CG of the whole system is lowered. In the case where the object weighs the same as an equal volume of water (same density), the CG of the system is unchanged whether the object rises or sinks. The object can be at any level beneath the surface without affecting the CG. You can see that a fish must weigh the same as an equal volume of water (have the same density); otherwise it would be unable to remain at different levels in the water. We will return to these ideas in Chapter 19, where liquids are treated in more detail. CHAPTER 11 ROTATIONAL EQUILIBRIUM

17 The CG of an iceberg is very far below the surface of the water it floats upon. Shake a box of stones of different sizes and observe what happens. The shaking enables the small stones to slip down into the spaces between the larger stones and in effect lower the CG. The larger stones therefore tend to rise to the top. The same thing happens when a tray of berries is gently shaken the larger berries tend to come to the top. You don t need to take a course in physics to know where to balance a baseball bat, how to stand a pencil upright on its flat end, or that you can t lean over and touch your toes if your heels are against a wall. With or without physics, everybody knows that it is easier to hang by your hands below a supporting rope than it is to stand on your hands above a supporting floor. And you don t need a formal study of physics to balance like a gymnast. But maybe it s nice to know that physics is at the root of many things you already know about. Knowing about things is not always the same as understanding things. Understanding begins with knowledge. So we begin by knowing about things, and then progress deeper to an understanding of things. That s where a knowledge of physics is very helpful. CONCEPT CHECK What happens to the center of gravity when an object is toppled? CONCEPT When an object is CHECK toppled, the center of gravity of that object is raised, lowered, or unchanged. 204 Teaching Resources Reading and Study Workbook Concept-Development Practice Book 11-3 PresentationEXPRESS Interactive Textbook Next-Time Questions 11-4, 11-5 Science, Technology, and Society CRITICAL THINKING Pseudoscience; they do not involve careful, repeated experimentation that produces consistent results. Science and Pseudoscience Science uses a powerful method of combining logic, observation, and experiment to find correlations, sometimes leading to a cause-and-effect relationship between things. It involves asking the kinds of questions science can handle, and searching for answers via careful, controlled experimentation. Only when repeated experiments produce consistent results and objective evidence is provided, is an idea scientifically valid. Such ideas reliably explain and predict many types of events. A pseudoscience is a false science. It claims the power of science to explain and predict events, but it is not based on the careful methods of science. Often, evidence cited by a pseudoscientist 204 Science, Technology, and Society to prove his or her case is subjective. Also, in pseudoscience, cause-and-effect relationships may be claimed, but no detailed logical connections can be provided. The danger of pseudoscience is that it can lead us to believe things that aren t true, or make us think we know things we don t. Thus, we may make unwise decisions. Nevertheless, pseudosciences appeal to many people. They can excite the imagination, simplify complex issues, and soothe anxiety about the unknown. Critical Thinking Are horoscopes that are seen frequently in newspapers and magazines an example of science or pseudoscience? Explain. How can you identify pseudoscience?

18 11 REVIEW For: Self-Assessment Visit: PHSchool.com Web Code: csa 1100 REVIEW Teaching Resources TeacherEXPRESS Conceptual Physics Alive! DVDs Center of Gravity Concept Summary To make an object turn or rotate, apply a torque. Key Terms torque (p. 189) lever arm (p. 190) center of mass (p. 192) center of gravity (p. 195) unstable equilibrium (p. 201) stable equilibrium (p. 201) neutral equilibrium (p. 201) When balanced torques act on an object, there is no change in rotation. The center of mass of an object is the point located at the object s average position of mass. For everyday objects, the center of gravity is the same as the center of mass. If the CG of an object is above the area of support, the object will remain upright. The CG of a person is not located in a fixed place, but depends on body orientation. When an object is toppled, the CG of that object is raised, lowered, or unchanged. think! Answers 11.1 No, because the lever arm is the same. To increase the lever arm, a better idea would be to use a pipe that extends upward In the center of the hole! No. A rigid object has one CG. If it is nonrigid, such as a piece of clay or putty, and is distorted into different shapes, then its CG may change as its shape is changed. Even then, it has one CG for any given shape You tend to hold your free arm outstretched to shift the CG of your body away from the load so your combined CG will more easily be above the base of support. To really help matters, divide the load in two if possible, and carry half in each hand. Or, carry the load on your head! CHAPTER CHAPTER ROTATIONAL ROTATIONAL EQUILIBRIUM EQUILIBRIUM

19 ASSESS Check Concepts 1. Force is a push that can accelerate objects; torque is force 3 lever arm that can rotate objects. 2. At a right angle to the lever arm 3. They have the same magnitude. 4. The heavier kid should sit closer to the fulcrum. 5. The center of mass 6. The center of mass is not at the midpoint because one end of the bat is more massive than the other end. 7. It should be kicked in line with its center of mass, not above or below it. 8. The center of mass follows a parabola before it explodes and the center of mass of the individual fragments follow the same parabola after the explosion. 9. They are the same for ordinary sized objects and different for large objects where gravity can vary. The moon s CG is slightly closer to Earth than its center of mass. 10. For a suspended object, the CG is below the point of suspension. For a standing object, the CG is above the support base. 11. Because its CG is above a support base 12. Until its CG extends beyond its support base 13. Your CG extends beyond your support base and a torque produces rotation. 14. So that their CG can pass beneath the bar; then the bar is higher than the actual jumping height! 11 ASSESS Check Concepts Section How does torque differ from force? 2. In what direction should a force be applied to produce maximum torque? Section How do clockwise and counterclockwise torques compare when a system is balanced? 4. For two kids of different masses balancing on a seesaw, should the heavier kid sit closer or farther from the fulcrum compared with the lighter kid? Section What part of an object follows a smooth path when the object is made to spin through the air or across a flat smooth surface? 6. Why is the center of mass of a baseball bat not at its midpoint? 7. To kick a football so that it doesn t rotate through the air, where should it be kicked relative to its center of mass? 8. Describe the motion of the center of mass of a fireworks projectile, before and after it explodes in midair. For: Visit: Web Code: Section When are the center of gravity and center of mass of an object the same? Give an example of when they can be different. 10. Where is the center of gravity of an object that hangs in equilibrium? For an object that stands in equilibrium? Section Why does the Leaning Tower of Pisa not topple? 12. How far can an object be tipped before it topples over? Section In terms of center of gravity, support base, and torque, why can you not stand with your heels and back to a wall and then bend over to touch your toes and return to your stand-up position? 14. Why do some high jumpers arch their bodies into a U shape when passing over the high bar? Section Distinguish between unstable, stable, and neutral equilibrium

20 11 ASSESS 16. Is the gravitational potential energy more, less, or unchanged when the CG of an object is raised? 17. What is the secret of balancing toys that exhibit stable equilibrium while appearing to be unstable? 18. What accounts for the stability of the Space Needle in Seattle? 19. If a container of dried beans with a table tennis ball at the bottom is shaken, what happens to the CG of the container? For: Visit: Web Code: 21. In a physics lab you find four different vertically mounted cart wheels that are not free to rotate. Each has a block that hangs from a string wrapped around the wheel. Rank the torques these blocks produce about the wheel axes from greatest to least. 15. Unstable: the CG is lowered with displacement; stable: the CG rises with displacement; neutral: the CG neither rises nor falls with displacement. 16. Potential energy is more when the CG is raised. 17. The CG of the toy hangs below the support point. 18. Its CG is below ground level so in order for it to topple its CG would have to be raised. 19. The CG would lower (and the container would become more stable). Think and Rank 20. D, A, C, B 21. C, D, A, B 22. A 5 B 5 C Think and Rank Rank each of the following sets of scenarios in order of the quantity or property involved. List them from left to right. If scenarios have equal rankings, then separate them with an equal sign. (e.g., A = B) 20. You hold a meterstick with the same suspended masses at the angles shown. Rank the torque needed to keep the stick steady from largest to smallest. 22. Perky (left) and Sneezlee (right) have the same mass and nicely balance at opposite ends of a seesaw. For the three positions, rank the length of the lever arm between Perky and the center of the seesaw from longest to shortest. A B C D 50 cm 75 cm CHAPTER CHAPTER ROTATIONAL ROTATIONAL EQUILIBRIUM EQUILIBRIUM