Physics Chapter 8 and up to 9.11 Notes
Physics Chapter 8 and up to 9.11 Notes PHYS 210
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Physics Chapter 8 Notes (Torque and Angular Momentum) Rotational Kinetic Energy and Rotational Inertia (8.1) To calculate kinetic energy of rotation, the speed of each particle is proportional to the angular speed of rotation, w. If an object consists of N particles, the sum of the kinetic energies of the particles can be written using the equation… o where the notation stands for the sum Q + Q + … + Q . 1 2 N The speed of each particle is related to its distance from the axis of rotation. o So, particles that are farther from the axis move faster, and closer move slower. o To find the speed of a particle moving in a circle V= r * w W= the angular speed R= the distance between the rotation axis and the particle To find the rotational kinetic energy, use the equation use: o Kinetic energy equations o Translational kinetic energy: o Rotational Kinetic energy: o Rotational Inertia: Torque (8.2) A quantity related to force You cannot exert a torque without exerting a force A measure of how effective a given force is at twisting or turning something. On something rotating around an axis, torque can change the rotational motion either by making it rotate faster or by slowing it down. Proportional to the magnitude of the force Only the tangential component of force produces a torque The tangential direction is perpendicular to both the radial direction and the axis of rotation; it is tangent to the circular path followed by a point on the object as the object rotates. Sign of torque indicates the direction of the angular acceleration that torque would cause by itself + (counter clockwise) (clockwise The sign of the torque is not determined by the sign of the angular velocity (in other words, whether the wheel is spinning CCW or CW); rather, it is determined by the sign of the angular acceleration the torque would cause if acting alone. To determine the sign of a torque, imagine which way the torque would make the object begin to spin if it is initially not rotating. To solve for torque: R= the distance between the rotation axis and the point of application of force F = the perpendicular component of force Units: N * m Calculating work done from the torque… (8.3) And to write work in the terms of torque… note that = rF and s = rΔ ; then Work is indeed the product of torque and the angular displacement. If and Δ have the same sign, the work done is positive; if they have opposite signs, the work done is negative. The Power, due to a constant torque, or the rate at which work is done is calculated using the equation… Rotational Equilibrium (8.4) if an object is also in rotational equilibrium, then the net torque acting on it must also be zero. Conditions for equilibrium : if the net force acting on an object is zero and the net torque about one rotation axis is zero, then the net torque about every other axis parallel to that axis must also be zero. Therefore, one torque equation is all we need. The best place to choose the axis is usually at the point of application of an unknown force so that the unknown force does not appear in the torque equation. To solve Equilibrium Problems, follow these steps… Identify an object or system in equilibrium. Draw a diagram showing all the forces acting on that object, each drawn at its point of application. Use the center of gravity as the point of application of any gravitational forces. To apply the force condition choose a convenient coordinate system and resolve each force into its x and ycomponents. To apply the torque condition ∑ = 0, choose a convenient rotation axis—generally one that passes through the point of application of an unknown force. Then find the torque due to each force. Use whichever method is easier: either the lever arm times the magnitude of the force or the distance times the perpendicular component of the force. Determine the direction of each torque; then either set the sum of all the torques (with their algebraic signs) equal to zero or set the magnitude of the CW torques equal to the magnitude of the CCW torques. Not all problems require all three equations (two force component equations and one torque equation). Sometimes it is easier to use more than one torque equation, with a different axis. Before diving in and writing down all the equations, think about which approach is the easiest and most direct. Equilibrium in the human body (8.5) A muscle has tendons at each end that connect it to two different bones across a joint (the flexible connection between the bones). When the muscle contracts, it pulls the tendons, which in turn pull on the bones. Thus, the muscle produces a pair of forces of equal magnitude, one acting on each of the two bones A muscle can pull but not push, so a flexor muscle such as the biceps cannot reverse its action to push the forearm away from the upper arm. The extensor muscles make bones move apart from each other. In the upper arm, an extensor muscle—the triceps—connects the scapula and humerus to the ulna (a bone in the forearm parallel to the radius) across the outside of the elbow. When the triceps contracts it pulls the forearm away from the upper arm. Using flexor and extensor muscles on opposite sides of the joint, the body can produce both positive and negative torques, although both muscles pull in the same direction. Rotational Form of Newtons Second Law (8.6) o Remember to assign the correct sign to each torque before adding them! o The sum of the torques due to internal forces acting on a rigid object is always zero. o Only include external torques o I= rotational inertia o A= rotational acceleration The angular acceleration of a rigid body is proportional to the net torque and inversely proportional to the rotational inertia. o More torque causes larger a, and more inertia causes a smaller a In rotational equilibrium the angular acceleration is zero, and the net torque is zero. The Motion of Rolling Objects (8.7) A rolling object combines translational motion of the center of mass with rotation about an axis that passes through the center of mass. For an object that is rolling without slippingCM = R. o There is a specific relationship between the rolling object's translational and rotational kinetic energies. The total kinetic energy of a rolling object is the sum of its translational and rotational kinetic energies. A wheel with mass M and radius R has a rotational inertia that is some pure number times MR ; it couldn't be anything else and still have the right units. 2 o We can write the rotational inertia about an axis through theCM as ICM= MR , o is a pure number that measures how far from the axis of rotation the mass is distributed. o Larger means the mass is, on average 2 Using ICM= MR and CM = R, the rotational kinetic energy for a rolling object can be written Since is the translational kinetic energy, This is convenient since depends only on the shape, not on the mass or radius of the object. For a given shape rolling without slipping, the ratio of its rotational to translational kinetic energy is always the same ( ). The total kinetic energy can be written or in terms of , Thus, two objects of the same mass rolling at the same translational speed do not necessarily have the same kinetic energy. The object with the larger value of has more rotational kinetic energy. Angular Momentum (8.8) Newton’s second law for translational momentum is written in two different ways: A more general form of Newton’s second law can be written as: o The net external torque acting on a system is equal to the rate of change of the angular momentum of the system To find angular momentum or the angular momentum of a rigid body rotating about a fixed axis… o Rotational inertia times the angular velocity Any change in angular momentum must be due to a change in angular velocity Conservation of Angular momentum If the net external torque acting on a system is zero, then the angular momentum of the system cannot change. o Li and Lf represent the angular momentum of the system at two different times. o Total energy, total linear momentum, and total angular momentum cannot change unless some external agent causes the change. o Conservation of angular momentum can be applied to any system if the net external torque on the system is zero. The conservation law refers to the total energy. By contrast, linear momentum and angular momentum cannot be added to find the “total momentum.” They are entirely different quantities, not two forms of the same quantity. They even have different dimensions, so it would be impossible to add them. Conservation of linear momentum and conservation of angular momentum are separate laws of physics. The Vector Nature of Angular Momentum (8.9) Torque and angular momentum are vector quantities. Angular momentum is conserved in both magnitude and direction in the absence of external torques. A special case is a symmetrical object rotating about an axis of symmetry. The direction of the angular momentum vector points along the axis of rotation. To choose between the two directions use the right hand rule. “Align your hand so that, as you curl your fingers in toward your palm, your fingertips follow the objects rotation. Then your thumb points in the direction of L. Chapter 9 Fluids (9.1) Solids, liquids, and gases Solids o Rigid o Not easily deformed by external forces o Molecules vibrate around fixed equilibrium positions o Do not have enough energy to break the bonds with neighbors Liquids o Do not hold their shapes o Does not have a definite shape o Incompressible – having a fixed volume that is impossible to change o The shape of the liquid can be changed by pouring it from a container of one shape into a container of a different shape, but the volume still stays the same. o Atoms almost as closely packed as those in the solid phase of the same material. o The intermolecular forces in a liquid are almost as strong as those in solids, but the molecules are not locked in fixed positions as they are in solids. Cold Water is one exception – the molecules are more closely packed than those in the solid phase. Gases o Cannot be characterized by a definite volume of shape. o Gas expands to fill its container and can easily be compressed. The molecules in a gas are very far apart. o Molecules are almost free of interactions, except when they collide. Pressure (9.2) A static fluid does not flow. Fluid pressure is caused by collisions of the fastmoving atoms or molecules of a fluid. A static fluid exerts a force on any surface with which it comes in contact; the direction of the force is perpendicular to the surface. A static fluid cannot exert a force parallel to the surface. o F – magnitude of the force acting perpendicularly to the surface. o A – area of the surface o P – pressure Pressure is a scalar quantity The force acting on an object in a submerged fluid, or on some portion of the fluid is vector and its direction is perpendicular to the surface. Pressure is the same anywhere in a fluid Notation – newtons per meters squared or pascal (Pa) Pascal’s Principle (9.3) If the weight of a static fluid is negligible (as, for example, in a hydraulic system under high pressure), then the pressure must be the same everywhere in the fluid. the fluid pressure must be the same everywhere in a weightless, static fluid. Pascal’s Principle o A change in pressure at any point on a confined fluid is transmitted everywhere throughout the fluid The Effect of Gravity on Fluid Pressure (9.4) Gravity makes fluid pressure increase as you move down and decrease as you move up. The density of a substance is its mass per unit volume. The Greek letter (rho) is used to represent density. The density of a uniform substance of mass m and volume V is Notation: kilograms per cubic meter Figuring out how pressure increases with depth due to gravity o Pressure variation with depth in a static fluid with uniform density o o Point 2 is a depth, d, below point 1 o This equation is to be applied to gases as long as the depth is small enough that changes in the density due to gravity are negligible. And to great depths in liquids o Pressure at a depth d below the surface of a liquid open to the atmosphere o Measuring Pressure (9.5) The manometer A mercury manometer consists of a vertical Ushaped tube, containing some mercury, with one side typically open to the atmosphere and the other connected to a vessel containing a gas whose pressure we want to measure o When both sides of the manometer are open to the atmosphere, the mercury levels are the same. o On the side where an object is connected, the gas will push the mercury down on the left side. Thus, the difference in mercury levels d is a measure of the pressure difference— commonly reported in millimeters of mercury (mm Hg). The pressure measured when one side of the manometer is open is the difference between atmospheric pressure and the gas pressure rather than the absolute pressure of the gas. This difference is called the gauge pressure, since it is what most gauges (not just manometers) measure: Some equations to solve for pressure in a manometer o o p is the density of mercury. o The difference in the pressures on the two sides of the manometer is o o Notation: mm Hg The Buoyant Force (9.6) The buoyant force is not a new kind of force exerted by a fluid, it is the sum of forces do to fluid pressure. o Fb – buoyant force o pV is the mass of the volume V of the fluid that the block displaces. o The buoyant force on the submerged block is equal to the weight of an equal volume of fluid. Archimedes’ Principle o A fluid exerts an upward buoyant force on a submerged object equal in magnitude to the weight of the volume of fluid displaced by the object. The net force due to gravity and buoyance acting on an object totally or partially immersed in a fluid is The force of gravity on an object The buoyant force is (Net force due to gravity) Specific Gravity is the ratio of its density to the density of water at 3.98 degrees Celsius. Fluid Flow (9.7) One difference between moving fluids and static fluids is that a moving fluid can exert a force parallel to any surface over or past which it flows; a static fluid cannot. Since the moving fluid exerts a force against a surface, the surface must also exert a force on the fluid. This viscous force opposes the flow of the fluid; it is the counterpart to the kinetic frictional force between solids. An external force must act on a viscous fluid (and thereby do work) to keep it flowing. Fluid flow can be characterized as steady or unsteady. When the flow is steady, the velocity of the fluid at any point is constant in time. The velocity is not necessarily the same everywhere, but at any particular point, the velocity of the fluid passing that point remains constant in time. The density and pressure at any point in a steadily flowing fluid are also constant in time. Steady flow is laminar. The fluid flows in neat layers so that each small portion of fluid that passes a particular point follows the same path as every other portion of fluid that passes the same point. The path that the fluid follows, starting from any point, is called a streamline (Fig. 9.19). The streamlines may curve and bend, but they cannot cross each other; if they did, the fluid would have to “decide” which way to go when it gets to such a point. The direction of the fluid velocity at any point must be tangent to the streamline passing through that point. When the fluid velocity at a given point changes, the flow is unsteady. Turbulence is an extreme example of unsteady flow (Fig. 9.20). In turbulent flow, swirling vortices— whirlpools of fluid—appear. The vortices are not stationary; they move with the fluid. The flow velocity at any point changes erratically; prediction of the direction or speed of fluid flow under turbulent conditions is difficult. The special case that we consider first is the flow of an ideal fluid. An ideal fluid is incompressible, undergoes laminar flow, and has no viscosity. Under some conditions, real fluids can be modeled as (nearly) ideal. The flow of an ideal fluid is governed by two principles: the continuity equation and Bernoulli's equation. The continuity equation is an expression of conservation of mass for an incompressible fluid: since no fluid is created or destroyed, the total mass of the fluid must be constant. Mass Flow Rate Volume Flow Rate Continuity equation for incompressible fluid Bernoulli’s Equation (9.8) This equation is a restatement of the principle of energy conservation applied to the flow of an ideal fluid. Some other useful equations Viscosity (9.9) Kinetic friction makes a sliding object slow down unless an applied force balances the force of friction. Similarly, viscous forces oppose the flow of a fluid. Steady flow of a viscous fluid requires an applied force to balance the viscous forces. The applied force is due to the pressure difference. Poiseuille’s Law o o where ΔV/Δt is the volume flow rate, ΔP is the pressure difference between the ends of the pipe, r and L are the inner radius and length of the pipe, respectively, and η is the viscosity of the fluid. the flow rate is inversely proportional to the viscosity of the fluid. The more viscous the fluid, the smaller the flow rate, if all other factors are equal. The volume flow rate ΔV/Δt for laminar flow of a viscous fluid through a horizontal, cylindrical pipe depends on several factors. First of all, the volume flow rate is proportional to the pressure drop per unit length(ΔP/L)—also called the pressure gradient. If a pressure drop ΔPmaintains a certain flow rate in a pipe of length L, then a similar pipe of length 2L needs twice the pressure drop to maintain the same flow rate (ΔP across the first half and another ΔP across the second half). Thus, the flow rate (ΔV/Δt) must be proportional to the pressure drop per unit length (ΔP/L) Viscous Drag (9.10) When an object moves through a fluid, the fluid exerts a drag force on it. When the relative velocity between the object and the fluid is low enough for the flow around the object to be laminar, the drag force derives from viscosity and is called viscous drag. The viscous drag force is proportional to the speed of the object (F D ). For larger relative speeds, the flow becomes turbulent and the drag force is proportional to the square of the object's speed (F D ). The viscous drag force depends also on the shape and size of the object. For a spherical object, the viscous drag force is given by Stokes's law: o where r is the radius of the sphere, η is the viscosity of the fluid, and is the speed of the object with respect to the fluid. Surface Tension (9.11) The surface of a liquid has special properties not associated with the interior of the liquid. The surface acts like a stretched membrane under tension. The surface tension (symbol , the Greek letter gamma) of a liquid is the force per unit length with which the surface pulls on its edge. The direction of the force is tangent to the surface at its edge. Surface tension is caused by the cohesive forces that pull the molecules toward each other. To show excess pressure o
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