Kinematics and Dynamics and Relativity Carlos Silva September

Kinematics and Dynamics and Relativity Carlos Silva September 30 th 2009

Isaac Newton (1643 -1727) Philosophie naturalis principia mathematica • Gravitational force • Motion laws • Principles of Mechanics: Motion of bodies • Developed Calculus Area of mathematics that deals with limits, derivatives, integrals Newton tree (? ) (Cambridge, UK)

Kinematics and Dynamics Kinematics • How to describe a motion of rigid bodies Linear motion Circular motion Dynamics • How forces affect motion of rigid bodies Force Torque Momentum conservation Newton’s laws of motion

Linear motion KINEMATICS

Time General definition (t) Clock reading (day, hour, minute, second) Physical definition (Δt) Elapsed time measured in seconds between two events Δ t = time at event B – time at event A (always positive)

Position, change in position and distance Position (x) Coordinates of a point in space Change in position (Δ x) Difference betwen coordinates of two different positions Δ x = x. B – x. A (signal gives the direction) Δx y A d Δy Distance Length of the path taken between position A and position B (always positive) Euclidean Distance (norm): D B x

Velocity and Acceleration Velocity Rate of position change: (derivative of position change in order to time) Acceleration Rate of velocity change: (derivative of velocity change in order to time) For constant acceleration and simple linear motions

Determine the motion Constant speed / null acceleration Motion with changing speed / constant acceleration Describe what type of motion did the object had in x:

Circular motion KINEMATICS

Angle, angular velocity and acceleration Angle (θ) Measured in radians (360º = 2π rad) Angular velocity (ω) Measured in rad s-1 Angular frequency is the magnitude of angular velocity Frequency (Hz=s-1) Number of events per second In this case is usual to measure in rpm (rotation per minute) Linear quantities of a particle Linear velocity Linear acceleration

Free fall objects motion DYNAMICS

Force Aristotle proposed “force” has the reason why an object puts another in motion Newton proposed that force is always a two object relation Gravity force is an interaction between Earth and another object Different forces may be acting on a single object Forces act at the distance Types of force Gravitational force Friction force (between two objects) Electrical Force Magnetic Force

Free fall motion Velocity graph of a falling object (experiment by Galileo) The acceleration is constant, regardless the mass! (9, 8 ms-2) This is the acceleration caused by Gravitation Force Gravitational acceleration g = 9, 8 ms-2 Gravity always “pulls” down Weight is the quantity of the force that attracts us to the ground (Weight in N, Mass in kg)

Projectile launch Object launched with horizontal speed It always fall due to gravity To achieve the longest distance, launch at 45º

Mass, Center of Mass, Inertia Mass Property of a body (how much matter does a body has) It becomes different form weight in places where the gravitational force is different from g (moon) Inertia All corps maintain their state of motion (rest or constant velocity) if no force is applied Center of Mass /Gravity Average of every position of a body weighted by their mass Point whose motion describes the object motion if all mass was concentrated in a single point Different from geometric center

Newton’s Laws First law If the sum of acting forces is zero, the center of mass continues in the same state of motion Second law If the acting forces are not zero, the acceleration of the body is proportional to the force Third law For each force, there is always an equal and opposite force

Momentum Conservative quantity of body (Ns) If no external force is acting on a body, the body maintains its momentum Product of mass by its velocity This explains several phenomena: Ballerina spinning

Torque (Moment of Force) Magnitude of the force applied to a rotational system (Nm) Equivalent to the Force on circular motion Power = torque x angular speed (Nms-1)

Friction Forces Static friction force is usually higher that kinetic friction force

Centripetal force (circular motion) Outward force (1) It doesn’t exist Inward force(2) Conservation of momentum (First Newton law) Centripetal force Force required to make a body follow a curve path

Kinetic and Potential Energy Kinetic Energy The work that it is necessary to bring an object from rest to the present velocity: Energy that a body possesses due to its motion: Potential Energy stored in a body that can be transformed into other type of energy Kinetic, thermal, chemical, elastic Gravitational potential energy Catapult

Examples DYNAMICS

Force acting on a spring Hooke’s law The deformation is proportional to the applied force that causes deformation Natural frequency Elastic Potential Energy

The pulley Allows to lift large masses into tall heights Gain mechanical advantage 2 pulleys, F/2 4 pulleys, F/4 Transfer mechanical forces across axes Motor Crane

The lever Based on the application of momentsforce Breaks Hand trucks Spring board Fishing rod What would be D 1 and D 2 in this case?

Ramp / inclined plane Reduce the force applied to lift at the expense of travelled distance Roman inclined plane (Masada, Israel)

Gears Transmits rotational forces between axes (like pulleys) Transforms rotational to linear motion

Crankshaft Transmits linear motion into rotational motion Crankset +pedal

Flyball governor Automatically controls the speed of the motor Regulating the fuel admission, based on the rotational acceleration Watt

Flywheels Used to store energy Significant inertia Used to attenuate peaks High energy density 130 W·h/kg, or ~ 500 k. J/kg Typical capacities range 3 k. Wh to 133 k. Wh Modern flywheel for storage (Beacon)

Pump Storage Used for load balancing Stores energy in the for of potential energy of water

Turbines (wind and water) Machine that extracts energy from a fluid flow Pelton wheel Wind turbine power Betz limit (59%) Horizontal, vertical

RELATIVITY

Relative velocity 3 ms-1 3 ms-1

Postulates of Relativity The Principle of Relativity (Galileo) • The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other. The Principle of Invariant Light Speed • Light in vacuum propagates with the speed c (a fixed constant) in terms of any system of inertial coordinates, regardless of the state of motion of the light source. Special principle of relativity: • If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K • Time dilation • Relativity of simultaneity • Velocity of light cannot be exceeded • Mass of an object near speed of light seems to increase • Equivalence of mass and energy (E=mc 2)

Twins paradox A twin who makes a journey into space in a high-speed rocket will return home to find he has aged less than his identical twin who stayed on Earth cms-1 0. 9 cms-1 0 ms-1