Motion in 2D and 3D Lecture

Motion in 2D and 3D: II ● Projectile Motion ○ Projectile: an object that is hit with force. moves in the x and y directions simultaneously. ■ Examples: balls hit in ball sports ○ Assume that: ■ Free fall acceleration is constant and directed downward ■ Earth is flat over the range of motion, and the range is relatively small compared to the radius of earth ■ Effect of air friction is negligible ■ Object follows a trajectory, or parabolic path ■ Important Diagram: ● Analyzing ○ ● position= (sum of initial position, position resulting from initial velocity, and position resulting from acceleration) ○ Initial velocity can be expressed in terms of components ■ vxi = vi cosθ and vyi = vi sinθ ○ Horizontal components (velocity in x direction) is constant. ax=0. ○ Y direction is free fall. ay=-g ○ Velocity is symmetrical. vi=-vf, and velocity 2 feet above from where projectile is fired is the same magnitude as velocity 2 feet before it returns to the starting height. ○ Y component of velocity, or the vertical velocity, is zero at maximum height, or halfway through total time Range and Maximum Height (assuming symmetrical motion) ○ Range: horizontal distance. ■ R= v0cosθ0t ■ Based on y= v0sinθ0t-½ag2, t= 2v0sinθ0/g ■ So, if we wished to eliminate time t, = se 2sinθcosθ=sin2θ) ○ ○ ○ ○ h: maximum height reached. Maximum range when theta is 45 degrees (°), because sin2θ is greatest when θ is 90 degrees. Maximum height when theta is 90 degrees, because the largest sinθ is sin90 degrees. Complementary angles produce the same range, because the motion is symmetrical. Height, eliminating time t t= x/(v0cosθ0) v0y= v0sinθ Motion in 2D and 3D ● Example (becau ○ ■ ■ ■ ■ Make a list of knowns and unknowns. ● vix= 55 m/s ● vi=55 m/s ● viy= 0 m/s ● h or y0= 500 m Solve for t ● Δy= viyt - ½ at2 ● -500= 0t - ½ (-9.8)t2 ● t= 10.1 s Solve for Δx ● Δx= vixt - ½ at2 ● = 55(10.1) - ½ (0)(10.12) ● = 555.5 m Solve for phi ● Ф= tan-1(Δx/Δh) ● = tan-1(555.5/500) ● = 48° degrees ○ ■ ■ Solve for vfy ● = viy + at ● = 0 - (-9.8)(10.1 s) ● = -99.0 m/s Solve for vf and θ ● vfx= 55 m/s ● vf=sqrt (vfx2+vfy2) ● vf= sqrt (552 = (-99)2) ● = 113 m/s ● θ= tan-1 (-99/55) ● = -60.9° ● ● ● Changing Velocity in Uniform Circular Motion ○ the change in the velocity vector of an object moving in a circle is due to the change in direction Centripetal Acceleration ○ Centripetal acceleration ac is always perpendicular to the path of motion ○ Always points toward the center of motion, so direction is always changing ● ○ Relative Velocity ○ Relies on a frame of reference described by a Cartesian coordinate system for which an observer is at rest with respect to the origin ○ ■ ■ ■ Reference frama SA is stationary Reference frame Sb is moving right relative to Sa at vAB These positions are related through velocity: ■ Galilean transformation equations: or Force and Motion I ● ● ● ● Force ○ ○ ○ Types ○ ○ According to Newton, is is what causes change in an object’s motion or position In other words, force causes an acceleration Measured in Newtons, or kg·m/s2. 1 N ~ ¼ lb Contact force: when 2 objects physically touch Field force: occur in empty space with no physical contact. ■ Fundamental forces: gravitational (between objects of different masses), electrical, nuclear (between subatomic particles), weak forces (radioactive decay) and magnetic forces Vector Nature ○ If forces are applied perpendicularly to each other, net force is the hypotenuse ○ Vector rules still apply for adding Newton’s First Law ○ Law of inertia: an object at rest tends to stay at rest, and in motion tends to stay in motion ○ If object doesn’t interact with other objects, or the net force is 0, there is a ● ● ● ● ● reference frame (inertial frame) in which the object has zero acceleration ■ Inertial frame is generally a frame where earth’s surface is stationary, or more accurately, moves with constant velocity relative to the distant stars ■ Any frame that moves with constant velocity relative to an inertial frame is itself an inertial frame ○ No net force → a=0, v=0 or constant ○ Inertia: tendency of object to resist attempts to change velocity. ○ Mass: property that specifies how much resistance an object exhibits to changes in its velocity. Greater mass=greater inertia ■ Scalar, independent of surroundings. inherent to object ○ Weight: magnitude of gravitational force exerted on object ■ Varies with location due to differences in gravitational forces ■ Ex: wearth = 180 lb; wmoon ~ 30 lb ■ mearth = 2 kg; mmoon = 2 kg Newton’s Second Law ○ Acceleration is directly proportional to net force and inversely proportional to mass ○ Net force= mass times net acceleration Fnet= manet ■ (when speed is much lower than the speed of light) ■ Works with components too ○ Gravitational force ■ Fg: force earth exerts on an object ■ Directed toward center of earth ■ Fg= mg= weight ■ g and weight are smaller at higher altitudes Newton’s Third Law ○ When 2 objects interact, F12 exerted by object 1 on object 2 is equal in magnitude but opposite in direction to F21 exerted by object 2 on object 1 ■ F12= - F21 ■ Every action has an equal and opposite reaction ■ Notation: FAB is the force exerted by A on B ■ These forces cannot be added, because they are on different objects ○ Normal force: reaction of force on object exerts back on the gravitational action force of the Earth. When the object is at rest and the surface is horizontal, it is equal in magnitude to weight, opposite direction Free Body Diagram: represents visually the forces acting on an object of interest in all directions Law Assumptions ○ Objects can be modeled as particles ○ Neglect reaction forces ○ Initially, we will deal with frictionless surfaces ○ Masses of strings or ropes are negligible ■ Force rope exerts is away from the object and parallel to the rope ■ When a rope attached to an object is pulling it, the magnitude of that force is the tension in the rope Equilibrium: acceleration is 0, or net force is 0 ● The cord, pulled taut, is under tension. If its mass is negligible, the cord pulls on the body and the hand with force T, even if the cord runs around a massless, frictionless pulley as in (b) and (c). Forces and Motion II ● Equilibrium ○ No acceleration ○ X and y components of force should add up to zero ● New Force ○ If there is acceleration, there must be a nonzero net force ○ Use free body diagrams and Newton’s second law (F= ma) in component form ○ First, determine direction of a ● Inclined Plane ○ Normal force is perpendicular to the plane, but the force of gravity is straight down ○ When making coordinate system of free body diagram, x should be parallel to the incline, and y should be perpendicular to the incline/parallel to the normal force, and convert Fg to its components in relation to this new coordinate system ○ Acceleration is always gsinθ ● Multiple Objects ○ Calculate the force of gravity for each object separately ● Force of Friction ○ Motion is resisted due to interactions between object and surface ○ Static friction is the force opposing an object at rest from going into motion, while dynamic friction opposes an object already moving from continuing to move Forces and Motion VI- Circular Motion and Other Applications ● Banked Curve ○ Horizontal circle ○ Designed with friction equaling zero ○ Component of normal force supplies the centripetal force ■ *vL2/rg Angle is independent of the mass If car rounds the curve slower than design speed, friction is necessary to keep it from sliding down the bank ○ If car goes faster, friction is necessary to keep it from sliding up Vertical Circle with Non-Uniform Speed ○ Gravitational force exerts tangential force on object ○ Tension at any point can be found Tension ○ at the bottom is a maximum ○ ○ ● ● ○ ● ○ If Ttop= 0, then vtop= Universal Gravitation Law ○ Fg only equals mg on Earth’s surface ○ Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the distance between them ○ Fg= G(m1m2/r2) ■ G, universal gravitational constant: 6.673x10-11 Nm2/kg2 ○ This is an inverse square law. The magnitude of the force varies as the inverse square of the sepration of the particles ○ ● ● at top is a minimum Vector form: ■ Negative sign means attractive force G vs g ○ G is the universal constant, the same everywhere ○ g is the acceleration due to gravity, 9.80 m/s2 at the surface of the Earth. Varies by location ■ g= GMEm/RE2 g above Earth’s surface ○ If object is some distance h above the earth’s surface, r becomes RE + h ○ g= (GME)/ ((RE+h)2) ○ So g decreases with increasing altitude ● ● ● ● ○ As r → infinity, weight of object approaches zero Kepler’s Laws ○ 1st: all planets move in elliptical orbits with the Sun at one focus ○ 2nd: the radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals ○ 3rd: square of the orbital period of any planet is proportional to the cube of the semimajor axis of elliptical orbit ■ Can be predicted from inverse square law ■ Assume circular orbit ■ Gravitational force supplies centripetal force T represents period of the motion Geosynchronous Satellite ○ appear s to remain over the same point on the earth ○ Grav force supplies a centripetal force Centrifugal Force ○ Fictitious force: results from an accelerated frame of reference. Appears to act on an object in the same way as a real force, but you can’t identify a second object for the fictitious force. It can’t be a real force because real forces are always interactions between 2 objects ○ Centrifugal force: outward force due to the centripetal acceleration associated with object’s change in direction Motion with Resistive Forces ○ Motion can be through a medium ■ Either a liquid or a fas ○ Medium exerts a resistive force, R, on an object moving through the medium ■ Magnitude of R depends on the medium ■ Direction is opposite the direction of motion of the object relative to the medium ■ Nearly always increases with increasing speed ■ Proportional to velocity ○ When R =mg, acceleration approaches 0, and v approaches a terminal velocity ● ● ● ● ● ○ When a bat and ball collide Intro to energy: ○ Energy/transfers/transformations involved in every physical process in the universe ○ Energy: scalar quantity; ability of one of more objects to do work ■ Can be transformed from one type to another, or transferred from one object to another ■ Total amount of energy is always the same- energy is conferred Types of Energy ○ Kinetic energy: associated with movement of members of a system. ○ Potential energy: stored energy. determined by the configuration of the system. ■ Sub-types include gravitational and elastic ○ Internal energy: relates to heat/temperature of system Types of Systems ○ Non-isolated systems ■ Energy can cross system boundary in a variety of ways, so… ■ Total energy of system can change ○ Isolated Systems ■ Energy is kept contained within system ■ Total energy, therefore, is constant Energy Transfer ○ Work: transfers energy by applying a force and causing a displacement of the point of application of the force ○ Mechanical waves: allow a disturbance to propagate through a medium; transfer energy ○ Heat: is driven by a temperature difference between 2 regions in space ○ Matter transfer: matter physicalls crosses the boundary of the system; direct ○ Electrical transmission: transfer is by electric current ○ Electromagnetic radiation: energy is transferred by electromagnetic waves ■ All energy is derived from the sun ○ ● ● ● Energy approach: sometimes it is hard or impossible to use Newton’s laws, we sue the system approach and stop looking at situations in particle form System: a small portion of the universe ○ A single particle, a collection of objects, a region of space, etc Kinetic energy: energy of particle due to motion. scalar. ○ k= ½ mv2. k= kinetic energy, m= mass, v= velocity ○ Change in kinetic energy is a possible result of doing work to transfer energy into a system ○ ● Work (W) done on a system by an agent exerting constant force F = (magnitude of F)*(magnitude of Δr, the displacement of the point of application of the force)*(cosθ, where θ is the angle between force and displacement vectors) ○ W= FΔrcosθ ○ When θ= 90°, like with normal force and gravitational force, cosθ= 0 and no work is done ○ We multiply F by cosθ because force, displacement, and therefore work must be all in the same plane/axis ● Units of Work ○ A scalar ● ● ● ○ Unit is a joule (J), which = Newton * meter = kg*m2/s2 More ○ Remember to determine what exactly is doing the work and what is it acting upon ○ Work is negative when force is in opposite direction of displacement (i.e. tension pulling upward something that is moving downward), and positive when they’re in same direction (i.e. gravity pulling object downward) ○ When work is done positively, it transfers energy; if work is negative, energy is transferred from system Work by Varying Force ○ When looking at a force vs displacement graph, total work= area under curve ○ Total work= sum of individual forces algebraically Work Done by a Spring ○ Object on a spring is a model of common physical system for which force varies with position ○ ○ If object is on horizontal, frictionless surface, pulling or pushing the spring to displace it from its equilibrium position will cause it to oscillate forever ○ Hooke’s Law ■ Simple harmonic motion ■ Force exerted by spring (Fs)= -kx ■ k= spring or force constant; measures stiffness of spring. Bigger as stiffness increases. ■ x= position of block with respect to equilibrium position, which is x=0 ■ work= ½ kx2 ○ Work-Kinetic Energy Theorem ■ Work done by net force= change in kinetic energy ■ vf= sqrt(2Fd/m) or sqrt(2W/m) ■ Normal and gravitational forces do no work when object is moving horizontally since they are perpendicular to direction of displacement ○ Power: time rate of energy transfer ■ power= W/t when the method of energy transfer is work ■ Measured is watts, or J/s ○ Potential Energy ■ Potential energy: stored energy ■ Change in potential energy (ΔPE or ΔU)= negative work done by gravity or elastic ■ Change in gravitational potential energy (ΔPEg or ΔUg)= mgΔh ■ Elastic potential energy: PEe or Us= ½ kx2 ● Can be converted to kinetic energy ● Always positive ● ● ● ● ● ● ● ● ● Internal Energy ○ Energy associate with object’s temperature ○ Can be converted from kinetic energy Conservative vs non-conservative forces ○ Conservative force: a force that can do the same amount of work for a given displacement, regardless of what the exact path the thing it does work on takes ■ Ex. gravity; whether a ball is thrown up from a cliff or the ball is dropped, the work is the same ○ Nonconservative force: a force whose work depends on the exact path the object takes ■ Ex. friction; the longer the path, the more work friction does Energy Diagrams and Stable Equilibrium ○ Equilibrium when force= 0 ○ Stable equilibrium when Ux is 0 at equilibrium, unstable equilibrium when Ux isn’t 0 at equilibrium ○ Neutral equilibrium: when U is constant over some region Free Fall ○ K+U is constant b/c of law of conservation of energy Momentum: mass times velocity ○ vector ○ Always conserved in a system Impulse-momentum theorem: impulse= change in momentum Impulse: vector quantity. Area under force vs time curve. Measured in same units as momentum Impulse approximation: though there may be multiple forces acting, in problems, assume force exerted that is being analyzed is very large compared with other forces Collisions ○ Collision: when 2 particles come close to each other and interact by means of forces. Usually, but not always, uses physical contact ○ Time interval of change in velocity is assumed to be short ○ Interaction forces assumed to be much greater than any other external forces ○ Types: ■ Elastic collision: momentum and kinetic energy are conserved ● Perfectly elastic collision: at the miscroscopic level ● Imperfectly elastic collision: a tiny bit of energy is lost to deformation, etc. ● ● Solving most problems would require 2 equations. This can be used instead: v1i – v2i = v1f + v2f ● ● ■ Special occasions: ○ Very heavy particle collides with very tiny one, and the havy one’s vellocity is unchanged while the tiny one gets 2x as fast ○ Very light psrticle Inelastic collision: KE is not conserved (objects change speed), but momentum is conserved. ● Perfectly inelastic collision: When the 2 objects stick together after colliding. ● ● ● ● ● Rigid Object Model ○ Rigid object: non-deformable. ■ Relative locations of all particles making up object stay constant ■ Model useful for problems where you want deformation to be negligible, like when you want to analyze the motion of an extended object Angular Position ○ Angular position: angle θ between reference line on object and fixed reference line in space. Object’s position in terms of an angle ○ Axis of rotation: center of disc ○ Object is at point P, a fixed distance r from the origin ■ Since r is fixed, as particle moves, only coordinate that moves is θ ■ As particle moves, it moves through an arc of length s. ● s= θr. θ must be measured in radians for rotational equations. 1 radian= 180°/pi= 180°/π Angular displacement: angle through which particle rotates during some time interval. Change in θ Average angular speed (omega, ω): ratio of angular displacement to time interval. Δθ/Δt ● ● Angular acceleration (alpha, α) ○ Δω/Δt ○ Measured in rad/s2 or s-2 Linear distance, and in turn velocity and acceleration is based on distance in meters rather than the change in angle. ○ Direction of linear velocity and acceleration, or tangential velocity and acceleration, is tangent to circular path. ● ● Directions ○ Calculating speed and acceleration will only give magnitude ○ Right hand rule gives directions: imagine holding a bar around which the object rotates. If the object rotates counterclockwise, speed/acceleration are positive and the thumb points upward, in the direction. For clockwise, it is the opposite ● Rotational Kinematics ○ Rigid object under constant angular acceleration: when acceleration is unchanging/ ○ Uses the same equations used in kinematics when acceleration is constant ● Relationships ○ Distance or arc length (s) = θ times radius ® ○ Linear velocity v= ωr ○ Linear acceleration= αr ○ Angular speed and acceleration are constant, but linear speed and acceleration change ○ ● Centripetal acceleration: ac= v2/r= rω2 ● Total acceleration (due to change in magnitude and speed) ○ ● Rotational Kinetic Energy ○ Ki = ½ mivi2 substitute ωir for vi ○ Total rotational kinetic energy ■ ○ Moment of inertia: l represents moment of inertia.proportionality factor of torque and angular acceleration ■ ■ Measured in kg*m2 ■ ● Torque: Ftr= mr2α. Tendency of a force to rotate an object about some axis. ● Parallel-Axis Theorem ○ I = ICM + MD 2 ○ ● ● ● ● ● ● Damped oscillations ○ When nonconservative forces are present, system is non-ideal ○ Example of damped motion is when object is attached to spring and submerged in viscous liquid ○ Amplitude of motion (A) decreases over time ○ Equations ■ Resistive force R = -bv, where b is a constant called damping coefficient ■ Restoring force of -kx; Fnet= -kx + -bv = ma ■ x= Ae-(b/2m)tcos(ωt+Φ {phi}) ■ Angular frequency ω= sqrt(k/m - (b/2m)2=sqrt(ωi2 - (b/2m)2 Types of Damping ○ Overdamped: b/2m<ωi ○ Underdamped: b/2m>ωi ○ Critically damped: b/2m=ωi, bvmax> kA. When they system stops oscillating ○ Forced Oscillations ○ After driving force on initially stationary object begins to act, amplitude of oscillation will increase ○ Amplitude of driven oscillation: A= Fo/m/sqrt((ωf2-ωi2)2+(bω/m)2 ■ ωi is natural or resonance frequency of system Resonance ○ Driving frequency= natural frequency Elasticity ○ How things would deform beyond rigid body approximation ○ Stress: force applied per unit of area ○ Strain: degree of deformation. Change in L/Li ○ Elastic modulus: stress divided by strain, constant or proportionality ■ Young’s Modulus: measures resistance of solid to a change in its length ■ Shear: measures resistance of motion of planes within a solid parallel to each other ■ Bulk: measures desistance due to change in volume ■ Pressure is stress ■ When modulus is constant, object is behaving elastically