Introductory Physics (week 3) @K302 Yasuyuki Matsuda
Today s Contents Velocity and Acceleration Newton s Laws of Motion
Position, Velocity, Acceleration
Particle Particle : An point-like object with its mass Has no volume, no shape, no structure Result of idealization/abstraction/approximation of a (real) object surrounding us. As a first step towards understand our world, We first describe the motion of a particle with laws of physics. Then we describe the motion of an object (which has volume), later in the course.
Motion of a particle The aim: We want to know the position of the particle as a function of time. Important quantities in the motion of a particle are Velocity Acceleration
Position vector To locate a particle, we need to Define a reference point (the origin) Define the distance from the origin Define the direction from the origin Definition Vector : A quantity which has its magnitude and direction the magnitude is always positive real number Scalar : A quantity which is represented by a single real number It can be positive or negative
Position vector To locate a particle, we need to Define a reference point (the origin) Define the distance from the origin : the magnitude Define the direction from the origin : the direction of the position vector of the particle Definition Vector : A quantity which has its magnitude and direction the magnitude is always positive real number Scalar : A quantity which is represented by a single real number It can be positive or negative
Velocity (1-dim) Velocity of a moving particle P O x 1 x 2 x This is the mean velocity over the time interval t 1 and t 2
Example x the average velocity between t 1 and t 2 is x 2 = x (t 2 ) displacement x 1 = x(t 1 ) this value is the slope of the green line O t 1 t 2 time t
Example x x (t 1 + t) average velocity during x (t 1 ) O t 1 t 1 + t t
Example x average velocity during average velocity during x (t 1 + t) x (t 1 ) O t 1 t 1 + t t
Example x average velocity during average velocity during average velocity during x(t 1 + t) x (t 1 ) O t 1 t 1 + t t
Example x at the limit of t 0 Instantaneous velocity at t = t 1 derivative x (t 1 ) is the slope of the tangent line average velocity during average velocity during average velocity during x(t 1 ) O t 1 t
Velocity (1-dim) Velocity of a moving particle P O P1 P2 x This is the mean velocity over the time interval t 1 and t 2 Definition: This can be positive (moving towards the direction increasing x) or negative.
Velocity (2- and 3-dim) When a particle moves in 2-dim or 3-dim space, its position is described by its displacement vector from an origin O. is a function of time O
Velocity (2- and 3-dim) When a particle moves in 2-dim or 3-dim space, its position is described by its displacement vector from an origin O. is a function of time O
Velocity (2- and 3-dim) When a particle moves in 2-dim or 3-dim space, its position is described by its displacement vector from an origin O. is a function of time O
Velocity (2- and 3-dim) When a particle moves in 2-dim or 3-dim space, its position is described by its displacement vector from an origin O. is a function of time The velocity vector is always tangential to the path the tangent line of the path O
Velocity (2- and 3-dim) When a particle moves in 2-dim or 3-dim space, its position is described by its displacement vector from an origin O. is a function of time Definition: The velocity is a vector quantity (it has its magnitude and its direction). The magnitude of velocity is called speed.
Differentiation of scalar & vector functions The scalar f is a function of the scalar variable t, f(t). The derivative of the function f(x) with respect to t is The vector is a function of the scalar variable t,. The derivative of the function with respect to t is
Differentiation rules for vector functions Rules Formula is often used.
Acceleration The rate at which the velocity changes Definition
Acceleration In a linear (1-dim) motion In a linear motion, the acceleration vector is parallel to the velocity vector.
Acceleration In a uniform circular motion (circular motion with a constant speed) In a uniform circular motion, the acceleration vector is perpendicular to the velocity vector. (We will prove it later).
Acceleration In general three-dimentional motion of a particle the tangent line of the path The acceleration vector has a component which is tangential to the path, and a component which is normal to the path. cf. The velocity vector is always tangential to the path
Memo: Newton s notation A derivative of a function with respect to time (time derivative) is often denoted using over-dot. This is called Newton s notation. First derivative Second derivative This notation o is only used for time derivatives. ves.
Differentiation in Cartesian Coordinate System When we use the Cartesian Coordinate System This simplicity is one of the reasons for which we often This simplicity is one of the reasons for which we often use the Cartesian Coordinate System.
Example: constant acceleration in linear motion A particle is moving with a constant acceleration on a straight line. O x
Example: constant acceleration in linear motion A particle is moving with a constant acceleration on a straight line. O x integration
Example: constant acceleration in linear motion A particle is moving with a constant acceleration on a straight line. O x integration integration
Example: constant acceleration in linear motion A particle is moving with a constant acceleration on a straight line. O x integration integration
Example: constant acceleration in linear motion When a particle is moving on a straight line with a constant acceleration, following equations hold true. Some of you have learned the equations above as formulas to solve the problem dealing with a linear motion with a constant acceleration. Now we have proved the formula mathematically. So we don t have to remember the formula what we need to remember is the way of thinking in which we derived the formula.
Reference Frames Reference Frame : a coordinate system on which the motion of a particle is described. When we have two frames P r If Frame 2 does not rotate relative to Frame 1, r O Frame 1 D O Frame 2 the velocity of P observed in Frame 1 the velocity of P observed in Frame 2 the velocity of Frame 2 observed in Frame 1 This is the so-called velocityaddition rule.
Reference Frames Reference Frame : a coordinate system on which the motion of a particle is described. P r If Frame 2 does not rotate relative to Frame 1,. Differentiate t it leads to. r D O Frame 2 Especially when Frame 2 is moving with a constant velocity, O Frame 1
Summary of position, velocity, and acceleration The position of a particle is described by position vector. The velocity of the particle is defined as The velocity vector is tangential to the path of the particle. The acceleration of the particle is defined as The acceleration vector has a component tangential to the path, and a component normal to the path.
Newton s Laws of Motion Newton s Laws of Motion and the law of gravitation
Newton s Laws of Motion Principia (1687) by Issac Newton First Law When all external influences on a particle are removed, the particle moves with constant velocity. Second Law When a force acts on a particle of mass, the particle moves with acceleration a given by the formula Third Law When two particles exert forces upon each other, these forces are equal in magnitude, opposite in direction, and parallel to the straight line joining the two particles.
Inertial Frame In what frame of reference are the laws true? Definition : Inertial Frame is a reference frame in which the First Law is true. If there exists one inertial frame, all frames moving with constant velocity y( (and no rotation) relative to the one are inertial frame. The First Law is a statement that there exist inertial frames.
Note: The Newton s Laws of Motion are laws of physics they are founded d upon experimental evidence, and stand or fail according to the accuracy of their predictions. Einstein s s special relativity (1905) When we describe the motion of a particle, exact inertial frames are not available. But practically, we can take a reference frame and consider it as an inertial frame. Motion of a ball, a pendulum Frame attached to the Earth Rotation of the Earth is normally a small correction, which is often negligible. Motion of a satellite around the Earth the geocentric frame Motion of a planet the heliocentric frame
Question: Are these are inertial frames? Suppose the frame attached to the earth is a inertial frame, The frame attached to a bicycle running down a slope without applying a brake? No. It is accelerating relative to the reference frame. The frame attached to the floor of a train whose speed is constant? Yes. The frame attached to the floor of a carousel whose rotating speed is constant? No. It is rotating.
Inertial mass Second Law When a force F acts on a particle of mass m, the particle moves with acceleration a given by the formula Definition : Inertial mass of a particle is a numerical measure of the reluctance of the particle to be accelerated.
The third law (action-reaction law) When two particles exert forces upon each other, these forces are equal in magnitude, opposite in direction, i and parallel l to the straight line joining the two particles. F 12 particle 1 F 21 particle 2
The third law mass m A gravitational force mg, which is proportional to the mass is exerted on a dropping apple. Where is the reaction to this force? The force with same magnitude, with opposite direction is exerted on the Earth.
The third law The forces acting on the apple The earth exerts downward force mg The desk exerts upward force with same magnitude mg The apple does not move, because it is in a state of equilibrium. The forces acting on the desk The apple exerts downward force mg The Earth exerts upward force mg The desk does not move, because it is in a state of equilibrium. The forces acting on the Earth The apple exerts upward force mg The desk exerts downward force mg The earth does not move, because it is in a state of equilibrium.
The second law : the Equation of Motion Given the force F acting the particle P, the position r of P is determined by the equation of motion F determines, not r In order to determine r as a function of t, we need to solve differential equation.
The law of Gravitation The gravitational force that two particles exert upon each other have magnitude where M and m are the particle masses, R is the distance between the particles. G is the constant of gravitation. The value of the constant of gravitation G=6.674 10 11 Nm 2 /kg 2 must be determined by experiments. No explanation why this constant should have this value.
Measurement of G Cavendish experiment (1797~1798) Using a torsion balance, he measured the attraction force (gravitational force) between lead balls. large ball: 158kg, small ball 0.73kg, length of the rod 1.8m
Gravitational force by spheres The gravitational force exerted by a symmetric sphere of The gravitational force exerted by a symmetric sphere of mass M is exactly the same as if the sphere were replaced by a particle of mass M located at the center.
The gravitation acceleration g two particles are under gravitational attraction force exerted by a particle of mass M The induced acceleration of particle is independent of the particle s mass
The gravitation acceleration g When we consider the motion of a particle under the gravity of the Earth, the motion often has an extent that is insignificant compared to the Earth s radius. In this case, it is often appropriate to approximate that the particle s acceleration is a constant vertical (downward) vector. This is called uniform gravity.
The gravitation acceleration g
Measurement of G Cavendish experiment (1797~1798) Using a torsion balance, he measured the attraction force (gravitational force) between lead balls. This was actually the first measurement of the mass of This was actually the first measurement of the mass of the Earth!
Gravitational mass Newton s law of gravity states the mass acts as a source of gravitational force. Definition: Gravitational mass of a particle is a property of the particle to determine the gravitational force.
Inertial mass Second Law When a force acts on a particle of mass m, the particle moves with acceleration a given by the formula Definition : Inertial mass of a particle is a numerical measure of the reluctance of the particle to be accelerated.
The principle of equivalence The proposition that inertial mass and gravitational mass are exactly equal is called the principle of equivalence. There have been many experiments to detect any difference between inertial mass and gravitational mass. The legend says that Galileo released different masses from the top of the Leaning Tower of Pisa, and found that they reached the ground at the same time. This is an experiment to test the principle of equivalence! Th l t t i t th diff i l th 10 11 The latest experiment says the difference is less than 10-11 (one in 100 billion), if any.