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Notes: Torque and Static Equilibrium Sections 7.8, 8.1, 8.4, 8.5, 9.1, 9.2, & 9.4 A circle is just a straight line rolled up. Rotational Quantities If you spin a wheel, and look at how fast a point on the wheel is spinning, the answer depends on how far away the point is from the axle. Velocity, then, isn't the most convenient thing to use when you're dealing with rotation, and for the same reason neither is displacement, or acceleration; it is often more convenient to use their rotational equivalents. The equivalent variables for rotation are angular displacement (θ)(angle, for short); angular velocity (ω), and angular acceleration (α). All the angular variables are related to the straight-line variables by a factor of r, the distance from the center of rotation to the point you're interested in. Although points at different distances from the center of a rotating wheel have different velocities, they all have the same angular velocity. Angles (angular displacements, that is) are generally measured in radians, and therefore all other angular quantities are expressed in radians as well. Any equation dealing with rotation can be found from its straight-line motion equivalent by substituting the corresponding rotational variables. The rotational kinematics equations apply when the angular acceleration is constant. The equations are the same as the constant-acceleration equations for 1-D motion, substituting the rotational equivalents of the straight-line motion variables. v = vo + at ………………….. ________________ d = vot + ½ at2 ……………....________________ d = ½ (v + vo) ……………… ________________ v2 = vo2 + 2ad ……………… ________________ Center of Gravity For a given body, the center of mass is the average location of all the mass that makes up the object. A symmetrical object like a ball can be thought of as having all of its mass concentrated at its geometric center; by contrast, an irregularly shaped object such as a baseball bat has more of its mass toward one end. A solid cone has its center of mass exactly one fourth of the way up from its base. The center of gravity of an object is the point you can suspend the object from without there being any rotation because of the force of gravity, no matter how the object is oriented. If you suspend an object from any point, let it go and allow it to come to rest, the center of gravity will lie along a vertical line that passes through the point of suspension. Center of gravity is the same thing as the center of mass, except specifically referring to an object under the influence of gravity. The terms are effectively synonymous, and we will use the abbreviation CG for short, when consideration of this position is necessary. The CG of a uniform but elogated object, such as a meter stick, is at its geometric center, for the stick acts as though its entire weight were concentrated there. Support at that single point supports the whole stick. Balancing an object provides a simple method of locating its CG. Excerpted from http://physics.bu.edu/~duffy/py105.html 147036790 The CG of any freely suspended object lies directly beneath (or at) the point of support. If a vertical line is drawn through the point of suspension, the CG lies somewhere along that line. To determine exactly where it lies along the line, we have only to suspend the object from some other point and draw a second vertical line through that point of suspension. The CG lies where the two lines intersect. The CG may be a point where no matter exists. For example, the center of mass of a ring or a hollow sphere is at the geometrical center. For any object, the x-position of the center of gravity can be calculated by considering the weights and xpositions of all the pieces making up the object: A similar equation would allow you to find the y position of the center of gravity. Fact 1 - An object thrown through the air may spin and rotate, but its center of gravity will follow a smooth parabolic path, just like a ball. Fact 2 - If you tilt an object, it will fall over only when the CG lies outside the supporting base of the object. Fact 3 - If you suspend an object so that its center of gravity lies below the point of suspension, it will be stable. It may oscillate, but it won't fall over. Torque Translational motion is the movement of a particle along a path, where the position of the force on the particle does not affect the motion of the particle. Rotational motion considers rigid bodies rather than particles, where the position of the applied force has a large effect on the rotation of the object. In the diagram to the right, F1 might cause some translational motion of the beam but probably little rotation. F2 might cause some translational motion of the beam but more likely, the beam will rotate. What would happen if F2 were moved closer to the end of the beam? Would it beam easier or harder to rotate? F2 F1 Torque is the measure of the tendency of a force to rotate a body around some point, called the fulcrum or pivot. The force is applied in such a way that it could cause rotation of an object if it is not balanced. τ= Fr sinθ Mathematically, it is defined as the product of the applied force and the perpendicular distance from some point (typically measured from the fulcrum). This length measurement is often called the "lever arm” or “moment arm". Only the perpendicular component of the force effects the rotation of the lever. Torque is maximum when r is large, and sinθ = 1, so θ is 90º. The units for torque are usually Nm (NOT the same as a Joule!) Direction of torque is usually described as clockwise or counterclockwise with respect to the fulcrum. The Second Condition of Equilibrium A net torque, Στ, is analogous to a net force (ΣF) when examining its influence on motion. Στ applies to rotational motion the same way ΣF applies to linear motion. Excerpted from http://physics.bu.edu/~duffy/py105.html 147036790 If something is at equilibrium, that means that in addition to ΣF = 0, there is the condition that Στ = 0. All of the torques that cause the object to rotate clockwise must be balanced by all of the torques that cause the object to rotate counterclockwise. Στcw + Στccw = 0. This means that the clockwise torques are equal to the counterclockwise torques, Στcw = Στccw. A few important notes about torque: Any torque that would try to cause counterclockwise rotation is considered to be positive. The exact location of where a force acts now becomes very important. The weight of an object seems to act through its center of mass. The words "balanced" or "equilibrium" or "at rest" imply Στ = 0. An object at equilibrium has no net force acting on it, and has no net torque acting on it. To solve Torque problems: Draw a sketch. Label the object with given information. Label the weight of the object as a force at the center of gravity (generally the exact middle of the object). Choose a fulcrum. NOTE: If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque; the location of the axis is completely arbitrary. Show the direction of clockwise motion with an arrow. Write the equation, Στcw = Στccw. Write the sum of the clockwise torques, as F1·r1+ F2·r2 + F3·r3, etc…. Repeat for counterclockwise torques. Substitute in any numbers that you have. Solve for the unknown. Rotational Inertia (moment of inertia) Just as an object at rest tends to stay at rest, and an object in motion tends to remain moving in a straight line, an object rotating about an axis tends to remain ROTATING about the same axis unless interfered with by some external influence. The property of an object to resist changes in rotation is called rotational INERTIA. Things that rotate tend to remain rotating, while non-rotating things tend to remain non-rotating. Symbol: I, kgm2 Like linear inertia, rotational inertia of an object depends on its MASS. The greater the mass the GREATER the rotational inertia. Unlike linear inertia, rotational inertia of an object depends on the POSITION of the mass. The greater the distance between the bulk of an object’s mass and its axis of rotation, the GREATER the rotational inertia. In general, the moment of inertia of a rigid body will be given in any problem you might be asked to answer. Note: The units of moment of inertia are kgm2. Excerpted from http://physics.bu.edu/~duffy/py105.html 147036790 Note: The complete set of dynamical equations needed to describe the motion of a rigid body consists of the torque equation given above, plus Newton's Second Law applied to the center of mass of the object: The moment of inertia, like torque, must be defined about a particular axis. It is different for different choices of axes. Extended objects can again be considered as a very large collection of much smaller masses glued together to which the definition of moment of inertia given above can be applied. Examples of Moments of Inertia of Extended Objects: uniform hoop: I = mr 2 cylindrical shell I = (1/12) mr 2 long thin rod (about middle) I = mL 2 long thin rod (about one end) I =⅓ mL 2 solid cylinder I = ½ mr 2 solid sphere I = (2/5)mr 2 The moment of inertia depends on how the mass is DISTRIBUTED about the axis. For a given total mass, the moment of inertia is greater if more mass is FARTHER from the axis. An object where the mass is concentrated close to the axis of rotation is EASIER to spin. Rotational Kinetic Energy A spinning object has rotational kinetic energy. The kinetic energy of rotation of a rigid body is again analogous to the translational KE. A rolling object has both translational and rotational kinetic energy. Remember: The law of conservation of energy holds in all situations. Angular Momentum Rotating things, whether a colony in space, a cylinder rolling down an incline, or an acrobat doing a somersault, keep on rotating until something stops rotation. A rotating object has an “inertia of rotation”. This is called Angular Momentum. It depends upon Rotational Inertia and rotational velocity. The faster something is rotating, the more angular momentum it has. The harder it was to get it started rotating (the greater its inertia), the more angular momentum it has. A planet orbiting the sun, a rock whirling at the end of a string, and the tiny electrons whirling about atomic nuclei all have angular momentum. Just as an external net force is required to change the inertia of an object, an external net torque is required to change the angular momentum of an object. Angular momentum is conserved if no net torque acts on the system. This means that if the rotational inertia increases, then the rotational velocity must decrease, (or vice versa) in order to maintain a constant angular momentum. Because angular momentum is proportional to the moment of inertia, which depends on not just the mass of a spinning object, but also on how that mass is distributed relative to the axis of rotation, some interesting effects can be observed. Excerpted from http://physics.bu.edu/~duffy/py105.html 147036790 Questions and Problems: 1. Where is the CG of the earth’s atmosphere? 2. Why is it dangerous to slide open the top drawers of a fully loaded file cabinet or dresser that is not secured to the floor or wall? 3. When a car drives off a cliff, why does it rotate forward as it falls? 4. Why doesn’t the Tower of Pisa fall over? 5. How can you design objects to reduce the likelihood of tipping? 6. If you wish to have maximum speed a the very bottom of a roller coaster ride, should you sit in a front car, a middle car, or a rear car? 7. A weight of 2 N is placed 0.2 m from the pivot of a 0.5-N beam. If the beam is 1-m long and the pivot is in the exact center, where should you place a 1.5 N weight to balance the beam? 8. A weight of 2 N is placed 0.2 m from the pivot of a 0.5-N beam. If the beam is 1-m long and the pivot is in the exact center, how much weight should be placed at 0.4 m from the pivot to balance the beam? 9. A weight of 2 N is placed 0.2 m from the pivot of a 0.5-N beam. If the beam is 1-m long and the pivot is at the 0.3-m mark, where should you place a 1.5 N weight to balance the beam? 10. A uniform bridge span weighs 50 x 103 N and is 40.0 m long. An automobile weighing 15 x 103 N is parked with its center of gravity located 12.0 m from the right pier. What upward support force is provided by the left pier? 11. A child wants to use a 10 kg board that is 3.5 m long as a seesaw. Since all her friends are busy, she balances the board by putting the support 1-m away from her when she sits at one end. What is her mass? 12. A uniform board weighs 500 N and is 10.0 m long. It overhangs a building roof, extending over the edge of the roof by 2.5 m. A paint bucket filled with sand, weighing 150 N, is sitting at the end of the board. How far out is it safe for the 700 N worker to walk on the board? Assuming he can stretch no more than 0.5 m, can he reach the bucket when he needs it? 13. A 1.4-kg rod is supported by a single rope at an angle of 34° over the rod to its connection point on the wall. The rod is attached to the wall on the other side by a hinge. Assume that the rod is uniform. (a) What is the tension in the rope? (b) What are the two components of the support force exerted by the hinge? 14. Which is easiest to rotate—when most of the mass is close to the axis or far from the axis? 15. Consider balancing a hammer upright on the tip of your finger. If the head of the hammer is heavy and the handle long, would it be easier to balance with the end of the handle on your fingertip so that the head is at the top, or the other way around with the head at your fingertip and the end of the handle at the top? 16. Consider a pair of meter sticks standing nearly upright against a wall. If you release them, they’ll rotate to the floor in the same time. But what if one has a massive hunk of clay stuck to its top end? Will it rotate to the floor in a longer or shorter time? 17. The Earth moves about the Sun in an elliptical orbit. As the Earth moves closer to the Sun, does the Earth-Sun system's moment of inertia increase, decrease, or remain constant? 18. Two hoops or rings (I = MR2) are centered, lying on a phonograph record. The smaller one has a radius of 0.05 m and the larger a radius of 0.1 m. Both have a mass of 3 kg. What is the total moment of inertia as the record turns around? Ignore the mass of the record. 19. A majorette takes two batons and fastens them together in the middle at right angles to make an "x" shape. Each baton was 0.8 m long and each ball on the end is 0.20 kg. (Ignore the mass of the rods.) What is the moment of inertia if the arrangement is spun around an axis through the center perpendicular to both rods? 20. A uniform 10-m-long , 50-N ladder rests against a smooth vertical wall. If the ladder is just on the verge of slipping when the angle it makes with the ground is 50º, find the coefficient of static friction between the ladder and ground? Excerpted from http://physics.bu.edu/~duffy/py105.html 147036790 21. A woman who weighs 500 N is standing on a board that weighs 100 N. The board is supported at each end, and the support force at the right end is 3 times bigger than the support force at the left end. If the board is 8 m long, how far from the right end is the woman standing? 22. A uniform 40-N board supports 2 children weighing500 N and 350 N. The support is under the center of gravity of the board, and the 500-N child is 1.50 m from the center. A) Determine the upward force, N, exerted on the board from the support. B) Determine where the 350-N child should sit to balance the system. 23. A uniform, horizontal, 300-N beam, 5.00-m long, is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53.0 with the horixontal. If a 600-N person stands 1.50-m from the wall, find (a) the tension in the cable and (b) the force exerted on the beam by the wall. 24. You've climbed up to the top of a 7.5 m high telephone pole. Just as you reach the top, the pole breaks at the base. Are you better off letting go of the pole and falling straight down, or sitting on top of the pole and falling down to the ground on a circular path? Or does it make no difference? 25. The total kinetic energy of a baseball thrown with a spinning motion is a function of which of the following? a. its linear velocity but not rotational velocity b. its rotational velocity but not linear velocity c. both linear and rotational velocities d. neither linear nor rotational velocity 26. Our galaxy may have begun as a huge cloud of gas and particles. Suppose the original cloud was far larger than the present size of the galaxy, was more or less spherical, and was rotating very much more slowly than at present. Gravitation between particles would have pulled them closer. What would be the role of angular momentum conservation on the galaxy’s shape and present rotational speed? 27. A broom balances at its center of gravity. If you saw the broom into 2 parts through the center of gravity and then weigh each part on a scale, which part will weigh more? 28. A skater, starting a spin with their arms extended, quickly pulls her arms in close to the body. What effect does this have on the skater? Why? 29. The Earth's gravity exerts no torque on a satellite orbiting the Earth in an elliptical orbit. Compare the motion at the point nearest the Earth (perigee) to the motion at the point farthest from the Earth (apogee). At the point closest to the Earth a. the angular velocity will be greatest although the linear speed will be the same. b. the speed will be greatest although the angular velocity will be the same. c. the kinetic energy and angular momentum will both be greater. d. none of the above. 30. A 40 kg boy is standing on the edge of a stationary 30 kg platform that is free to rotate. The boy tries to walk around the platform in a counterclockwise direction. As he does a. the platform doesn't rotate. b. the platform rotates in a clockwise direction just fast enough so that the boy remains stationary relative to the ground. c. the platform rotates in a clockwise direction while the boy goes around in a counterclockwise direction relative to the ground. d. both go around with equal angular velocities but in opposite directions. 31. A figure skater on ice with arms extended, spins at a rate of 2.0 rev/s. After she draws her arms in, she spins at 5 rev/s. By what factor does her moment of inertia change in the process? Excerpted from http://physics.bu.edu/~duffy/py105.html 147036790