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In two and three dimensions, the acceleration vector can have an arbitrary direction and does not necessarily point along a given component of the velocity.
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The instantaneous acceleration is produced by a change in velocity taken over a very short (infinitesimal) time period. Instantaneous acceleration is a vector in two or three dimensions. It is found by taking the derivative of the velocity function with respect to time.
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In three dimensions, accelerationa→(t)a→(t)can be written as a vector sum of the one-dimensional accelerationsax(t),ay(t),andaz(t)ax(t),ay(t),andaz(t)along thex-,y-, andz-axes.
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The kinematic equations for constant acceleration can be written as the vector sum of the constant acceleration equations in thex,y, andzdirections.
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Projectile motion is the motion of an object subject only to the acceleration of gravity, where the acceleration is constant, as near the surface of Earth.
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To solve projectile motion problems, we analyze the motion of the projectile in the horizontal and vertical directions using the one-dimensional kinematic equations forxandy.
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The time of flight of a projectile launched with initial vertical velocityv0yv0yon an even surface is given byTtof=2(v0sinθ)g.Ttof=2(v0sinθ)g.This equation is valid only when the projectile lands at the same elevation from which it was launched.
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The maximum horizontal distance traveled by a projectile is called the range. Again, the equation for range is valid only when the projectile lands at the same elevation from which it was launched.
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Uniform circular motion is motion in a circle at constant speed.
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Centripetal accelerationa→Ca→Cis the acceleration a particle must have to follow a circular path. Centripetal acceleration always points toward the center of rotation and has magnitudeaC=v2/r.aC=v2/r.
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Nonuniform circular motion occurs when there is tangential acceleration of an object executing circular motion such that the speed of the object is changing. This acceleration is called tangential accelerationa→T.a→T.The magnitude of tangential acceleration is the time rate of change of the magnitude of the velocit...
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An object executing uniform circular motion can be described with equations of motion. The position vector of the object isr→(t)=Acosωti^+Asinωtj^,r→(t)=Acosωti^+Asinωtj^,whereAis the magnitude|r→(t)|,|r→(t)|,which is also the radius of the circle, andωωis the angular frequency.
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When analyzing motion of an object, the reference frame in terms of position, velocity, and acceleration needs to be specified.
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Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies with the choice of reference frame.
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IfSandS′S′are two reference frames moving relative to each other at a constant velocity, then the velocity of an object relative toSis equal to its velocity relative toS′S′plus the velocity ofS′S′relative toS.
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If two reference frames are moving relative to each other at a constant velocity, then the accelerations of an object as observed in both reference frames are equal.
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F → net = ∑ F → = F → 1 + F → 2 + ⋯ F → net = ∑ F → = F → 1 + F → 2 + ⋯
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v → = constant when F → net = 0 → N v → = constant when F → net = 0 → N
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F → net = ∑ F → = m a → F → net = ∑ F → = m a →
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F net = m a F net = m a
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∑ F → x = m a → x , ∑ F → y = m a → y , and ∑ F → z = m a → z . ∑ F → x = m a → x , ∑ F → y = m a → y , and ∑ F → z = m a → z .
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F → net = d p → d t F → net = d p → d t
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w → = m g → w → = m g →
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w = m g w = m g
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F → AB = − F → BA F → AB = − F → BA
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N → = − m g → N → = − m g →
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N = m g N = m g
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N = m g cos θ N = m g cos θ
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F → = − k x → . F → = − k x → .
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T = w = m g T = w = m g
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dynamics : study of how forces affect the motion of objects and systems
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external force : force acting on an object or system that originates outside of the object or system
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force : push or pull on an object with a specific magnitude and direction; can be represented by vectors or expressed as a multiple of a standard force
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free fall : situation in which the only force acting on an object is gravity
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free-body diagram : sketch showing all external forces acting on an object or system; the system is represented by a single isolated point, and the forces are represented by vectors extending outward from that point
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Hooke’s law : in a spring, a restoring force proportional to and in the opposite direction of the imposed displacement
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inertia : ability of an object to resist changes in its motion
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inertial reference frame : reference frame moving at constant velocity relative to an inertial frame is also inertial; a reference frame accelerating relative to an inertial frame is not inertial
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law of inertia : see Newton’s first law of motion
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net external force : vector sum of all external forces acting on an object or system; causes a mass to accelerate
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newton : SI unit of force; 1 N is the force needed to accelerate an object with a mass of 1 kg at a rate of1m/s21m/s2
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Newton’s first law of motion : body at rest remains at rest or, if in motion, remains in motion at constant velocity unless acted on by a net external force; also known as the law of inertia
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Newton’s second law of motion : acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system and is inversely proportional to its mass
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Newton’s third law of motion : whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts
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normal force : force supporting the weight of an object, or a load, that is perpendicular to the surface of contact between the load and its support; the surface applies this force to an object to support the weight of the object
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tension : pulling force that acts along a stretched flexible connector, such as a rope or cable
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thrust : reaction force that pushes a body forward in response to a backward force
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weight : forcew→w→due to gravity acting on an object of massm
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Dynamics is the study of how forces affect the motion of objects, whereas kinematics simply describes the way objects move.
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Force is a push or pull that can be defined in terms of various standards, and it is a vector that has both magnitude and direction.
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External forces are any outside forces that act on a body. A free-body diagram is a drawing of all external forces acting on a body.
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The SI unit of force is the newton (N).
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According to Newton’s first law, there must be a cause for any change in velocity (a change in either magnitude or direction) to occur. This law is also known as the law of inertia.
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Friction is an external force that causes an object to slow down.
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Inertia is the tendency of an object to remain at rest or remain in motion. Inertia is related to an object’s mass.
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If an object’s velocity relative to a given frame is constant, then the frame is inertial. This means that for an inertial reference frame, Newton’s first law is valid.
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Equilibrium is achieved when the forces on a system are balanced.
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A net force of zero means that an object is either at rest or moving with constant velocity; that is, it is not accelerating.
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An external force acts on a system from outside the system, as opposed to internal forces, which act between components within the system.
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Newton’s second law of motion says that the net external force on an object with a certain mass is directly proportional to and in the same direction as the acceleration of the object.
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Newton’s second law can also describe net force as the instantaneous rate of change of momentum. Thus, a net external force causes nonzero acceleration.
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Mass is the quantity of matter in a substance.
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The weight of an object is the net force on a falling object, or its gravitational force. The object experiences acceleration due to gravity.
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Some upward resistance force from the air acts on all falling objects on Earth, so they can never truly be in free fall.
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Careful distinctions must be made between free fall and weightlessness using the definition of weight as force due to gravity acting on an object of a certain mass.
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Newton’s third law of motion represents a basic symmetry in nature, with an experienced force equal in magnitude and opposite in direction to an exerted force.
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Two equal and opposite forces do not cancel because they act on different systems.
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Action-reaction pairs include a swimmer pushing off a wall, helicopters creating lift by pushing air down, and an octopus propelling itself forward by ejecting water from its body. Rockets, airplanes, and cars are pushed forward by a thrust reaction force.
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Choosing a system is an important analytical step in understanding the physics of a problem and solving it.
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When an object rests on a surface, the surface applies a force to the object that supports the weight of the object. This supporting force acts perpendicular to and away from the surface. It is called a normal force.
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When an object rests on a nonaccelerating horizontal surface, the magnitude of the normal force is equal to the weight of the object.
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When an object rests on an inclined plane that makes an angleθθwith the horizontal surface, the weight of the object can be resolved into components that act perpendicular and parallel to the surface of the plane.
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The pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension. When a rope supports the weight of an object at rest, the tension in the rope is equal to the weight of the object. If the object is accelerating, tension is greater than weight, and if it is accelerating oppos...
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The force of friction is a force experienced by a moving object (or an object that has a tendency to move) parallel to the interface opposing the motion (or its tendency).
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The force developed in a spring obeys Hooke’s law, according to which its magnitude is proportional to the displacement and has a sense in the opposite direction of the displacement.
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Real forces have a physical origin, whereas fictitious forces occur because the observer is in an accelerating or noninertial frame of reference.
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To draw a free-body diagram, we draw the object of interest, draw all forces acting on that object, and resolve all force vectors intox- andy-components. We must draw a separate free-body diagram for each object in the problem.
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A free-body diagram is a useful means of describing and analyzing all the forces that act on a body to determine equilibrium according to Newton’s first law or acceleration according to Newton’s second law.
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f k = μ k N f k = μ k N
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F c = m v 2 r or F c = m r ω 2 F c = m v 2 r or F c = m r ω 2
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tan θ = v 2 r g tan θ = v 2 r g
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F D = 1 2 C ρ A v 2 F D = 1 2 C ρ A v 2
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F s = 6 π r η v F s = 6 π r η v
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banked curve : curve in a road that is sloping in a manner that helps a vehicle negotiate the curve
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centripetal force : any net force causing uniform circular motion
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Coriolis force : inertial force causing the apparent deflection of moving objects when viewed in a rotating frame of reference
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drag force : force that always opposes the motion of an object in a fluid; unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid
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friction : force that opposes relative motion or attempts at motion between systems in contact
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ideal banking : sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net external force on the vehicle equals the horizontal centripetal force in the absence of friction
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inertial force : force that has no physical origin
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kinetic friction : force that opposes the motion of two systems that are in contact and moving relative to each other
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noninertial frame of reference : accelerated frame of reference
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static friction : force that opposes the motion of two systems that are in contact and are not moving relative to each other
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terminal velocity : constant velocity achieved by a falling object, which occurs when the weight of the object is balanced by the upward drag force
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Newton’s laws of motion can be applied in numerous situations to solve motion problems.
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Some problems contain multiple force vectors acting in different directions on an object. Be sure to draw diagrams, resolve all force vectors into horizontal and vertical components, and draw a free-body diagram. Always analyze the direction in which an object accelerates so that you can determine whetherFnet=maFnet=ma...
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The normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating vertically, the normal force is less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force is always less than the full weight of the object.
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Some problems contain several physical quantities, such as forces, acceleration, velocity, or position. You can apply concepts from kinematics and dynamics to solve these problems.
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Friction is a contact force that opposes the motion or attempted motion between two systems. Simple friction is proportional to the normal forceNsupporting the two systems.
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The magnitude of static friction force between two materials stationary relative to each other is determined using the coefficient of static friction, which depends on both materials.
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