Translational Motion

Translational motion, forces, work, energy, and equilibrium in living systems Translational Motion (PHY) • Units and dimensions • • • o One dimension = magnitude of length or distance only o Two dimensions = length or distance on a 2D plane (xy coordinates) o Three dimensions = length or distance in 3D space (xyz coordinates) o Four dimensions = length or distance in 3D space at a given time (xyzt coordinates) Vectors, components o Scalar: without direction ▪ e.g. length, time, mass o Vector: with direction ▪ e.g. displacement, acceleration, force o Components: the portion of the vector in a given direction Vector addition o You can directly add vectors if they are in the same direction o To add vectors in different directions, you must add their x, y, and z components. The resulting components make up the added vector o The vector sum of all components of a vector equal to the vector itself o Operation involving a vector and a vector may or may not result in a vector (kinetic energy from the square of vector velocity results in scalar energy) o Operation involving a vector and a scalar always results in a vector o Operation involving a scalar and a scalar always results in a scalar Speed, velocity (average and instantaneous) o Speed: scalar, no direction, rate of change in distance o Velocity: vector, has direction, rate of change in displacement o Average speed: • o Average velocity: o Instantaneous speed is the speed at an instant (infinitesimal time interval) o Instantaneous velocity is velocity at an instant (infinitesimal time interval) o Instantaneous speed equals instantaneous velocity in magnitude o Instantaneous velocity has a direction, instantaneous speed does not o The direction of instantaneous velocity is tangent to the path at that point Acceleration o Acceleration is the rate of change in velocity o Average acceleration: ▪ Uniformly accelerated motion along a straight line ▪ If acceleration is constant and there is no change in direction, all of the following applies: ▪ The value of speed/velocity, distance/displacement are interchangeable in this case, just keep a mental note of the direction ▪ ▪ For Cartesian coordinates, take upward and rightward motion as positive; down and left as negative For free falls, take downward as positive Force (PHY) • Newton’s First Law, inertia o Law of inertia states: without an external force acting on an object, nothing will change about that object in terms of speed and direction o In the absence of an external force: ▪ Something at rest will remain at rest ▪ Something in motion will remain in motion with the same speed and direction ▪ Objects are “inert” to changes in speed and direction • Newton’s Second Law (F=ma) o The unit for force is the Newton. N = kg•m/s2 o Both force and acceleration are vectors because they have direction • Newton’s Third Law, forces equal and opposite o Every action has an equal and opposite reaction • Friction, static and kinetic o Friction is a force that is always in the direction to impede the sliding of surfaces. • o Static friction: Kinetic friction: o μ is the coefficient of friction and N is the normal force ▪ Friction is a vector, however its direction is always opposite to the motion of surface involved ▪ Coefficient static friction is always larger than the coefficient of kinetic friction ▪ Lubricants reduce friction because they change surface properties and reduce the coefficient of friction Center of mass o The center of mass is the average distance, weighted by mass o In a Cartesian coordinate, the center of mass is the point obtained by doing a weighted average for all the positions by their respective masses o You can set the point of reference anywhere and use relative coordinates Equilibrium (PHY) • Vector analysis of forces acting on a point object o Static vs dynamic equilibrium: static = no net force, no movement; dynamic = no net force, movement o In equilibrium: sum of all forces and torques acting on a system equal zero o If NOT in equilibrium: center of mass is accelerating translationally and/or its parts are accelerating rotationally • Torques, lever arms o Torque is the angular equivalent of force - it makes things rotate, have angular acceleration, change angular velocity and direction ▪ Convention is that positive torque makes things rotate counterclockwise and negative torque makes things rotate clockwise o Depends on: ▪ Component of force perpendicular to the position vector (vector extending outwards from center of mass) ▪ Distance between point where force is applied and point where rotation occurs (greater distance = greater torque) o Lever arm consists of a lever (rigid rod) and a fulcrum (where the center of rotation occurs) ▪ Torque is the same at all positions of the lever arms (both on the same side and on the other side of the fulcrum) o If you apply a force at a long distance from the fulcrum, you exert a greater force on a position closer to the fulcrum o The catch: you need to move the lever arm through a longer distance Work (PHY) • Work done by a constant force: W = Fd cos θ o Work = energy transfer for any reason other than a temperature difference • Mechanical advantage o Mechanical advantage = little input force (effort) → large output force o Using ramps can achieve mechanical advantage ▪ Double distance of ramp = halving force needed o Using the lever arm can achieve mechanical advantage ▪ Doubling length of lever arm = halving force needed o Using pulleys can achieve mechanical advantage ▪ Vectors of tension going up help to lift force up ▪ Work required to lift mass hasn’t changed, just force • Work Kinetic Energy Theorem o Work on an object can transform into kinetic energy ▪ When you push on an object, it will move: Fd = ½ mv2 ▪ When gravity does work on an object, it will move: Fweighth = mgh = ½ mv2 o Kinetic energy of an object can do work A moving object can slide against friction for a while before coming to a stop: ½ mv2 = Ffrictiond o Sign conventions: negative = energy going out of system, positive = energy going into system Conservative forces o If a force doesn’t dissipate heat, sound or light, then it is a conservative force o Work done by conservative forces are path independent o Conservative forces are associated with potential energy o For example, the force from a spring can be stored as spring potential energy o Gravitational force can be stored as gravitational potential energy o Electromagnetic forces are also conservative o Non-conservative includes frictional forces and human exertion. When friction acts on an object, the heat and sound released cannot be recovered. When you flex your arm, you lose heat that cannot be recovered (you cannot re-absorb the heat you lost) ▪ • Energy of Point Object Systems (PHY) • Kinetic Energy: KE = ½ mv2; units o Unit = Joule o Speed is more important than mass for the KE because speed is squared • Potential Energy o PE = mgh (gravitational, local) o PE = ½ kx2 (spring) • Conservation of energy o Two types of energy transfer: heat and work ▪ Heat: energy transferred due to change in temperature ▪ Work: energy transferred due to any other reason o Principles of conservation ▪ Energy OUT of a system = Energy coming IN ▪ Total energy BEFORE = Total energy AFTER ▪ Total energy of a system = Sum of all forms of energy in that system o Law of conservation of mechanical energy: when only conservative forces are acting, sum of mechanical energies remains constant • Power, units o Power is the rate of energy use OR work per unit time o P = Fv cosθ, Power in applying a F to an object causing it to move at speed v ▪ Cosθ ensures force is moving in direction of velocity o Unit = Watt, or Joule per second o 3 categories of systems: ▪ Open: can exchange mass and energy ▪ Closed: can exchange energy but not mass ▪ Isolated: can’t exchange energy or mass Periodic Motion (PHY) • Amplitude, frequency, phase o Amplitude (A): how high the peaks are or how low the troughs are, in meters ▪ The displacement is how far the wave vibrates / oscillates about its equilibrium (center) position ▪ The amplitude is the maximum displacement ▪ Amplitude is correlated with the total energy of the system in periodic motion. Larger amplitude = greater energy o Period (T): the time it takes for one cycle, in seconds ▪ T = 1/ƒ • o Frequency (ƒ): the rate, or how many cycles per second, Hertz (cycles per second) ▪ ƒ = 1/T ▪ Sometimes, frequency is in rpm (revolutions per minute) rpm = cycles per second x 60 o Angular frequency (ω): the rate, in how many radians per second ▪ ω = 2πƒ ▪ ω is also called angular velocity o Phase ▪ In phase: the waves are 0 or 2π radians (0 or 360˚) apart. The resulting amplitude (sum of the waves) is twice the original ▪ Completely out of phase: the waves are π radians (180˚) apart. The resulting amplitude is zero ▪ Out of phase: resulting amplitude is between 0 and twice the original Transverse and longitudinal waves: wavelength and propagation speed o Transverse wave: wave displacement is perpendicular to the direction of motion ▪ Light ▪ Electromagnetic radiation ▪ A standing wave by oscillating a string side ways. The speed for such a wave = √(string tension/mass per unit length of the string) ▪ Know that tense, light strings can produce faster transverse waves o Longitudinal wave: wave displacement is parallel to the direction of motion ▪ Sound ▪ Pressure wave ▪ Earthquakes