Chapter 9: Static and Torque PDF

Chapter 9: Static and Torque Introduction to Statics and Torque Newton’s second law states that net F = ma, and so the net external force is zero for all stationary objects and for all objects moving at constant velocity. There are forces acting, but they are balanced. That is, the...

Chapter 9: Static and Torque Introduction to Statics and Torque Newton’s second law states that net F = ma, and so the net external force is zero for all stationary objects and for all objects moving at constant velocity. There are forces acting, but they are balanced. That is, they are in equilibrium. Statics is the study of forces in equilibrium, a large group of situations that makes up a special case of Newton’s second law. The First Condition for Equilibrium The first condition for equilibrium is the net external force on the system must be zero. Expressed as an equation, this is simply 𝑛𝑒𝑡 𝐹 = 0 Note that if 𝑛𝑒𝑡 𝐹 is zero, then the net external force in any direction is zero. For example, the net external forces along the typical x- and y-axes are zero. This is written as, 𝑛𝑒𝑡 𝐹𝑥 = 0 𝑎𝑛𝑑 𝑛𝑒𝑡 𝐹𝑦 = 0  This motionless person is in static equilibrium. The forces acting on him add up to zero. This car is in dynamic equilibrium because it is moving at constant velocity. There are horizontal and vertical forces, but the net external force in any direction is zero. The applied force between the tires and the road is balanced by air friction, and the weight of the car is supported by the normal forces. The Second Condition for Equilibrium The second condition necessary to achieve equilibrium involves avoiding accelerated rotation (maintaining a constant angular velocity). A rotating body or system can be in equilibrium if its rate of rotation is constant and remains unchanged by the forces acting on it. To understand what factors, affect rotation, let us think about what happens when you open an ordinary door by rotating it on its hinges.  (a) Counterclockwise torque is produced by this force, which means that the door will rotate in a counterclockwise due to F. Note that r┴ is the perpendicular distance of the pivot from the line of action of the force. (b) A smaller counterclockwise torque is produced by a smaller force acting at the same distance from the hinges (the pivot point). (c) The same force as in (a) produces a smaller counterclockwise torque when applied at a smaller distance from the hinges  (d) The same force as in (a), but acting in the opposite direction, produces a clockwise torque. (e) A smaller counterclockwise torque is produced by the same magnitude force acting at the same point but in a different direction. Here,θ is less than 90o. (f) Torque is zero here since the force just pulls on the hinges, producing no rotation. In this case θ = 0; The magnitude, direction, and point of application of the force are incorporated into the definition of the physical quantity called torque. Torque is the rotational equivalent of a force. It is a measure of the effectiveness of a force in changing or accelerating a rotation.  The equation form, the magnitude of torque is defined to be 𝜏 = 𝑟𝐹 sin 𝜃 where τ (the Greek letter tau) is the symbol for torque, r is the distance from the pivot point to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the vector directed from the point of application to the pivot point. An alternative expression for torque is given in terms of the perpendicular lever arm r┴ which is defined as, r┴ = 𝑟𝑠𝑖𝑛θ So that, τ = r┴ F The SI unit of torque is newtons times meters, usually written as N.m.  The torque is always calculated with reference to some chosen pivot point. For the same applied force, a different choice for the location of the pivot will give you a different value for the torque, since both r and θ depend on the location of the pivot. Note that for rotation in a plane, torque has two possible directions. Torque is either clockwise or counterclockwise relative to the chosen pivot point. Now, the second condition necessary to achieve equilibrium is that the net external torque on a system must be zero. An external torque is one that is created by an external force. The point can be the physical pivot point of a system or any other point in space—but it must be the same point for all torques.  If the second Condition is satisfied for one choice of pivot point, it will also hold true for any other choice of pivot point in or out of the system of interest. The second condition necessary to achieve equilibrium is stated in equation form as net τ = 0. where 𝑛𝑒𝑡 means total. Torques, which are in opposite directions are assigned opposite signs. A common convention is to call counterclockwise (ccw) torques positive and clockwise (cw) torques negative. Two children balancing a seesaw satisfy both conditions for equilibrium. The lighter child sits farther from the pivot to create a torque equal in magnitude to that of the heavier child. 1. When opening a door, you push on it perpendicularly with a force of 55.0 N at a distance of 0.850m from the hinges. What torque are you exerting relative to the hinges? 2. When tightening a bolt, you push perpendicularly on a wrench with a force of 165 N at a distance of 0.140 m from the center of the bolt. How much torque are you exerting in newton × meters ? 3. A 50.0N seesaw supports two people who weigh 455N and 525N respectively. The fulcrum is under the center of gravity of board. The 525N person is 1.50m from center. A) where does another person should sit so that seesaw is balanced? B) Find the upward normal force exerted by fulcrum? 4. Two Childrens are balanced on seesaw of negligible mass. First child has a mass of 26kg and sits 1.60m from the pivot, a) If second child is of mass 32kg, how far is she from the pivot? B) What is the force exerted by the pivot? Stability It is one thing to have a system in equilibrium; it is quite another for it to be stable. The toy doll perched on the man’s hand in figure, for example, is not in stable equilibrium. There are three types of equilibrium: stable, unstable, and neutral. Figure presents a balanced system, such as the toy doll on the man’s hand, which has its center of gravity (cg) directly over the pivot, so that the torque of the total weight is zero. This is equivalent to having the torques of the individual parts balanced about the pivot point, in this case the hand.  A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite to the direction of the displacement. This pencil is in the condition of equilibrium. The net force on the pencil is zero and the total torque about any pivot is zero.  A system is in unstable equilibrium if, when displaced, it experiences a net force or torque in the same direction as the displacement from equilibrium. A system in unstable equilibrium accelerates away from its equilibrium position if displaced even slightly. An obvious example is a ball resting on top of a hill. Once displaced, it accelerates away from the crest.  A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position. A marble on a flat horizontal surface is an example. Combinations of these situations are possible. For example, a marble on a saddle is stable for displacements toward the front or back of the saddle and unstable for displacements to the side. Figure shows another example of neutral equilibrium.  (a) The center of gravity of an adult is above the hip joints (one of the main pivots in the body) and lies between two narrowly-separated feet. Like a pencil standing on its eraser, this person is in stable equilibrium in relation to sideways displacements, but relatively small displacements take his cg outside the base of support and make him unstable.  Humans are less stable relative to forward and backward displacements because the feet are not very long. Muscles are used extensively to balance the body in the front-to-back direction. (b) While bending in the manner shown, stability is increased by lowering the center of gravity. Stability is also increased if the base is expanded by placing the feet farther apart. Birds such as chickens have easier systems to control. The cg of a chicken lies below its hip joints and between its widely separated and broad feet. Even relatively large displacements of the chicken’s cg are stable and result in restoring forces and torques that return the cg to its equilibrium position with little effort on the chicken’s part.  Some birds, such as the flamingo, have balance systems that are almost as sophisticated as that of humans. Engineers and architects strive to achieve extremely stable equilibriums for buildings and other systems that must withstand wind, earthquakes, and other forces that displace them from equilibrium. Although the examples in this section emphasize gravitational forces, the basic conditions for equilibrium are the same for all types of forces. The net external force must be zero, and the net torque must also be zero. Applications of Statics, Including Problem-Solving Strategies Statics can be applied to a variety of situations, ranging from raising a drawbridge to bad posture and back strain. Since statics is a special case of Newton’s laws, both the general problem-solving strategies and the special strategies for Newton’s laws. Problem solving strategy The first step is to determine whether or not the system is in static equilibrium. This condition is always the case when the acceleration of the system is zero and accelerated rotation does not occur. It is particularly important to draw a free body diagram for the system of interest. Solve the problem by applying either or both of the conditions for equilibrium net F = 0 and net τ = 0. If the second condition is involved, choose the pivot point to simplify the solution. Any pivot point can be chosen, but the most useful ones cause torques by unknown forces to be zero. Always choose a convenient coordinate system for projecting forces. Check the solution to see if it is reasonable by examining the magnitude, direction, and units of the answer. Simple Machines Simple machines are devices that can be used to multiply or augment a force that we apply – often at the expense of a distance through which we apply the force. The word for “machine” comes from the Greek word meaning “to help make things easier.” Levers, gears, pulleys, wedges, and screws are some examples of machines. Energy is still conserved for these devices because a machine cannot do more work than the energy put into it.  However, machines can reduce the input force that is needed to perform the job. The ratio of output to input force magnitudes for any simple machine is called its mechanical advantage (MA). 𝐹𝑜 𝑀𝐴 = 𝐹𝑖  One of the simplest machines is the lever, which is a rigid bar pivoted at a fixed place called the fulcrum. Torques are involved in levers, since there is rotation about a pivot point. Distances from the physical pivot of the lever are crucial, and we can obtain a useful expression for the MA in terms of these distances. A nail puller is a lever with a large mechanical advantage. The external forces on the nail puller are represented by solid arrows.  The force that the nail puller applies to the nail (Fo) is not a force on the nail puller. The reaction force the nail exerts back on the puller (Fn) is an external force and is equal and opposite to Fo. The perpendicular lever arms of the input and output forces are li and lo. Hence, 𝑙𝑖 𝐹𝑖 = 𝑙𝑜 𝐹𝑜 Rearranging the above equations we get, 𝐹𝑜 𝑙𝑖 = 𝐹𝑖 𝑙𝑜  Two other types of levers that differ slightly from the nail puller are a wheelbarrow and a shovel, The wheelbarrow and shovel differ from the nail puller because both the input and output forces are on the same side of the pivot. (a) In the case of the wheelbarrow, the output force or load is between the pivot and the input force. The pivot is the wheel’s axle. Here, the output force is greater than the input force. Thus, a wheelbarrow enables you to lift much heavier loads than you could with your body alone. (b) In the case of the shovel, the input force is between the pivot and the load, but the input lever arm is shorter than the output lever arm. The pivot is at the handle held by the right hand. Here, the output force (supporting the shovel’s load) is less than the input force (from the hand nearest the load), because the input is exerted closer to the pivot than is the output.  Another very simple machine is the Inclined Plane. Pushing a cart up a plane is easier than lifting the same cart straight up to the top using a ladder, because the applied force is less. However, the work done in both cases (assuming the work done by friction is negligible) is the same. Inclined lanes or ramps were probably used during the construction of the Egyptian pyramids to move large blocks of stone to the top. A Crank is a lever that can be rotated about 360o its pivot, Such a machine may not look like a lever, but the physics of its actions remain the same. An ordinary Pulley has an MA of 1; it only changes the direction of the force and not its magnitude. 5. In the wheelbarrow, the load has a perpendicular lever arm of 7.50 cm, while the hands have a perpendicular lever arm of 1.02 m. (a) What upward force must you exert to support the wheelbarrow and its load if their combined mass is 45.0 kg? (b) What force does the wheelbarrow exert on the ground? Forces and Torques in Muscles and Joints Muscles, bones, and joints are some of the most interesting applications of statics. There are some surprises. Muscles, for example, exert far greater forces than we might think. Muscles can only contract, so they occur in pairs. In the arm, the biceps muscle is a flexor—that is, it closes the limb. The triceps muscle is an extensor that opens the limb. This configuration is typical of skeletal muscles, bones, and joints in humans and other vertebrates.  Most skeletal muscles exert much larger forces within the body than the limbs apply to the outside world. The reason is clear once we realize that most muscles are attached to bones via tendons close to joints, causing these systems to have mechanical advantages much less than one.  The back is considerably more complicated than the arm or leg, with various muscles and joints between vertebrae, all having mechanical advantages less than 1. Back muscles must, therefore, exert very large forces, which are borne by the spinal column. Discs crushed by mere exertion are very common. People adjust their stance to maintain balance. (a) A father carrying his son piggyback leans forward to position their overall cg above the base of support at his feet. (b) A student carrying a shoulder bag leans to the side to keep the overall cg over his feet. (c) Another student carrying a load of books in her arms leans backward for the same reason.  What are the benefits of having most skeletal muscles attached so close to joints? One advantage is speed because small muscle contractions can produce large movements of limbs in a short period of time. Other advantages are flexibility and agility, made possible by the large numbers of joints and the ranges over which they function. THANK YOU