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The work done by a force is zero if the displacement is either zero or perpendicular to the force.
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The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction.
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The net workWnetWnetis the work done by the net force acting on an object.
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Work done on an object transfers energy to the object.
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The translational kinetic energy of an object of massmmmoving at speedvvisKE=12mv2KE=12mv2.
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The work-energy theorem states that the net workWnetWneton a system changes its kinetic energy,Wnet=12mv2−12mv02Wnet=12mv2−12mv02.
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Work done against gravity in lifting an object becomes potential energy of the object-Earth system.
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The change in gravitational potential energy,ΔPEgΔPEg, isΔPEg=mghΔPEg=mgh, withhhbeing the increase in height andggthe acceleration due to gravity.
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The gravitational potential energy of an object near Earth’s surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy,ΔPEgΔPEg, have physical significance.
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As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so thatΔKE= −ΔPEgΔKE= −ΔPEg.
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A conservative force is one for which work depends only on the starting and ending points of a motion, not on the path taken.
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We can define potential energy(PE)(PE)for any conservative force, just as we definedPEgPEgfor the gravitational force.
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The potential energy of a spring isPEs=12kx2PEs=12kx2, wherekkis the spring’s force constant andxxis the displacement from its undeformed position.
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Mechanical energy is defined to beKE+PEKE+PEfor a conservative force.
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When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form,
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where i and f denote initial and final values. This is known as the conservation of mechanical energy.
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A nonconservative force is one for which work depends on the path.
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Friction is an example of a nonconservative force that changes mechanical energy into thermal energy.
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WorkWncWncdone by a nonconservative force changes the mechanical energy of a system. In equation form,Wnc=ΔKE+ΔPEWnc=ΔKE+ΔPEor, equivalently,KEi+PEi+Wnc=KEf+PEfKEi+PEi+Wnc=KEf+PEf.
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When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newtonâ€...
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The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.
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When all forms of energy are considered, conservation of energy is written in equation form asKEi+PEi+Wnc+OEi=KEf+PEf+OEfKEi+PEi+Wnc+OEi=KEf+PEf+OEf, whereOEOEis allother forms of energybesides mechanical energy.
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Commonly encountered forms of energy include electric energy, chemical energy, radiant energy, nuclear energy, and thermal energy.
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Energy is often utilized to do work, but it is not possible to convert all the energy of a system to work.
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The efficiencyEffEffof a machine or human is defined to beEff=WoutEinEff=WoutEin, whereWoutWoutis useful work output andEinEinis the energy consumed.
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Power is the rate at which work is done, or in equation form, for the average powerPPfor workWWdone over a timett,P=W/tP=W/t.
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The SI unit for power is the watt (W), where1 W=1 J/s1 W=1 J/s.
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The power of many devices such as electric motors is also often expressed in horsepower (hp), where1 hp=746 W1 hp=746 W.
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The human body converts energy stored in food into work, thermal energy, and/or chemical energy that is stored in fatty tissue.
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Therateat which the body uses food energy to sustain life and to do different activities is called the metabolic rate, and the corresponding rate when at rest is called the basal metabolic rate (BMR)
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The energy included in the basal metabolic rate is divided among various systems in the body, with the largest fraction going to the liver and spleen, and the brain coming next.
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About 75% of food calories are used to sustain basic body functions included in the basal metabolic rate.
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The energy consumption of people during various activities can be determined by measuring their oxygen use, because the digestive process is basically one of oxidizing food.
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The relative use of different fuels to provide energy has changed over the years, but fuel use is currently dominated by oil, although natural gas and solar contributions are increasing.
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Although non-renewable sources dominate, some countries meet a sizeable percentage of their electricity needs from renewable resources.
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The United States obtains only about 10% of its energy from renewable sources, mostly hydroelectric power.
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Economic well-being is dependent upon energy use, and in most countries higher standards of living, as measured by GDP (Gross Domestic Product) per capita, are matched by higher levels of energy consumption per capita.
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Even though, in accordance with the law of conservation of energy, energy can never be created or destroyed, energy that can be used to do work is always partly converted to less useful forms, such as waste heat to the environment, in all of our uses of energy for practical purposes.
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change in momentum : the difference between the final and initial momentum; the mass times the change in velocity
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conservation of momentum principle : when the net external force is zero, the total momentum of the system is conserved or constant
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elastic collision : a collision that also conserves internal kinetic energy
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impulse : the average net external force times the time it acts; equal to the change in momentum
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inelastic collision : a collision in which internal kinetic energy is not conserved
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internal kinetic energy : the sum of the kinetic energies of the objects in a system
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isolated system : a system in which the net external force is zero
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linear momentum : the product of mass and velocity
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perfectly inelastic collision : a collision in which the colliding objects stick together
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point masses : structureless particles with no rotation or spin
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quark : fundamental constituent of matter and an elementary particle
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second law of motion : physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes
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Linear momentum (momentumfor brevity) is defined as the product of a system’s mass multiplied by its velocity.
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In symbols, linear momentumppis defined to bep=mv,p=mv,wheremmis the mass of the system andvvis its velocity.
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The SI unit for momentum iskg·m/skg·m/s.
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Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
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In symbols, Newton’s second law of motion is defined to beFnet=ΔpΔt,Fnet=ΔpΔt,FnetFnetis the net external force,ΔpΔpis the change in momentum, andΔtΔtis the change time.
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Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts:Δp=FnetΔt.Δp=FnetΔt.
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Forces are usually not constant over a period of time.
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The conservation of momentum principle is writtenptot=constantptot=constantorptot=p′tot(isolated system),ptot=p′tot(isolated system),ptotptotis the initial total momentum andp′totp′totis the total momentum some time later.
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An isolated system is defined to be one for which the net external force is zeroFnet=0.Fnet=0.
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During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.
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Conservation of momentum applies only when the net external force is zero.
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The conservation of momentum principle is valid when considering systems of particles.
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An elastic collision is one that conserves internal kinetic energy.
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Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions.
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An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).
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A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision.
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Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations.
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The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with thexx-axis parallel to the velocity of the incoming particle.
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Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (thexx-axis), stated bym1v1=m1v′1cosθ1+m2v′2cosθ2m1v1=m1v′1cosθ1+m2v′2cosθ2and along the direction perpendicular to the initial direction (theyy-axis) stated by0=m1v′1y+m2vâ...
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The internal kinetic before and after the collision of two objects that have equal masses is12mv12=12mv′12+12mv′22+mv′1v′2cosθ1−θ2.12mv12=12mv′12+12mv′22+mv′1v′2cosθ1−θ2.
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Point masses are structureless particles that cannot spin.
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Newton’s third law of motion states that to every action, there is an equal and opposite reaction.
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Acceleration of a rocket isa=vemΔmΔt−ga=vemΔmΔt−g.
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A rocket’s acceleration depends on three main factors. They areThe greater the exhaust velocity of the gases, the greater the acceleration.The faster the rocket burns its fuel, the greater its acceleration.The smaller the rocket's mass, the greater the acceleration.
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The greater the exhaust velocity of the gases, the greater the acceleration.
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The faster the rocket burns its fuel, the greater its acceleration.
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The smaller the rocket's mass, the greater the acceleration.
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center of gravity : the point where the total weight of the body is assumed to be concentrated
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dynamic equilibrium : a state of equilibrium in which the net external force and torque on a system moving with constant velocity are zero
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mechanical advantage : the ratio of output to input forces for any simple machine
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neutral equilibrium : a state of equilibrium that is independent of a system’s displacements from its original position
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perpendicular lever arm : the shortest distance from the pivot point to the line along whichFFlies
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SI units of torque : newton times meters, usually written as N·m
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stable equilibrium : a system, when displaced, experiences a net force or torque in a direction opposite to the direction of the displacement
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static equilibrium : a state of equilibrium in which the net external force and torque acting on a system is zero
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static equilibrium : equilibrium in which the acceleration of the system is zero and accelerated rotation does not occur
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torque : turning or twisting effectiveness of a force
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unstable equilibrium : a system, when displaced, experiences a net force or torque in the same direction as the displacement from equilibrium
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Statics is the study of forces in equilibrium.
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Two conditions must be met to achieve equilibrium, which is defined to be motion without linear or rotational acceleration.
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The first condition necessary to achieve equilibrium is that the net external force on the system must be zero, so thatnetF=0netF=0.
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The second condition assures those torques are also balanced. Torque is the rotational equivalent of a force in producing a rotation and is defined to beτ=rFsinθτ=rFsinθwhereττis torque,rris the distance from the pivot point to the point where the force is applied,FFis the magnitude of the force, andθθis the an...
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whereττis torque,rris the distance from the pivot point to the point where the force is applied,FFis the magnitude of the force, andθθis the angle betweenFFand the vector directed from the point where the force acts to the pivot point. The perpendicular lever armr⊥r⊥is defined to be
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so that
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The perpendicular lever armr⊥r⊥is the shortest distance from the pivot point to the line along whichFFacts. The SI unit for torque is newton-meter(N·m)(N·m). The second condition necessary to achieve equilibrium is that the net external torque on a system must be zero:netτ=0netτ=0By convention, counterclockwise...
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By convention, counterclockwise torques are positive, and clockwise torques are negative.
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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 the direction of the displacement.
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A system is in unstable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in the same direction as the displacement from equilibrium.
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A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position.
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Statics can be applied to a variety of situations, ranging from raising a drawbridge to bad posture and back strain. We have discussed the problem-solving strategies specifically useful for statics. Statics is a special case of Newton’s laws, both the general problem-solving strategies and the special strategies for ...
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