Describing Motion. Newton Newton s Laws of Motion. Position Velocity. Acceleration. Key Concepts: Lecture 9

Transcription

1 Key Concepts: Lecture 9 Newton Newton s Laws of Motion More on Kepler s Laws Describing Motion Position Velocity Rate of change of position (speed & direction) 80 km/hr Acceleration 40 km/hr Rate of change of velocity (a) Change in speed but not direction Newton s Law of Universal Gravitation 2 examples of acceleration: (b) Change in direction but not speed Newton Only child, posthumous son of an illiterate yeoman born prematurely - sickly as child raised by maternal grandmother as a child he built clocks & sundials practical joker Trinity College, Cambridge University at 18 studied mathematics & astrology encouraged to study physics by Barrow University closed in 1665 due to plague Invented calculus, studied gravity, optics Barrow resigns & gives Newton his post at Cambridge Newton s Laws of Motion Law I: Law of Inertia A body at rest or in motion at a constant velocity along a straight line remains in that state of rest or motion unless acted on by a net outside force. Takes next logical step beyond Galileo s definition of inertia (tendency of a body to keep moving after all forces stop acting on it) Uniform motion is just as natural a state for a body as being at rest

2 Laws of Motion Law II - The Force Law The acceleration (a) due to an applied force (F) is in the same direction as the force & is proportional to the strength of the force & is inversely proportional to the object s mass (m) The units of force are chosen so the constant is 1. So we write a = F/m To have acceleration there must be a force Force & acceleration always work in the same direction Given the same force, a more massive object accelerates more slowly than a less massive one a F / m a = constant x F /m constant = 1 a = F/m We can write this as F = m a Force=Mass xacceleration Question? A ball is attached to a string and I spin it abound my head in a circle Is the ball accelerating? If it is accelerating what is the force? If the string were to break what path would the ball follow? Examples of the Second Law Friction Hockey puck on ice vs. on a street Impact of a bat on a baseball The bat imparts a force to the ball and sends it flying in the opposite direction Laws of Motion Law III - The Reaction Law For every applied force, there is an equal, but opposite force Forces always occur in pairs a force cannot be created in isolation - need at least two bodies acting against each other if gravity is a force it must act between bodies

3 Newton Figures Out Gravity Question? You push a cart and it moves but you do not appear to move. Why don t you move if there is an opposite and equal force pushing on you? He unified the force which makes an apple drop from a tree and the force which makes the moon orbit the earth Gravity causes all objects to attract one another He intuitively figured out that the force of gravity between two objects depends on only three things: The two masses of the objects: more massive objects gives a stronger attractive force The distance between the objects: moving objects further apart weakens the force This is true on size scales from a laboratory desk to groups of stars and galaxies Newton s Law of Universal Gravitation Demonstration: 2nd and 3rd of Newton s Laws 2nd Law: Force = mass x acceleration (F = m a) 3rd Law: When two bodies interact the forces they feel are opposite in direction and equal in strength. You push a cart and it moves but you do not appear to move. Why? Because friction couples you to the massive Earth, which recoils only a very tiny amount. If we remove friction, then we can see the two motions more clearly Skate boards for Thur? Force is proportional to the masses, m1 and m2. Smaller masses smaller force Force is inversely proportional to the square of the distance between the objects. Further apart weaker force. This kind of dependence of force with distance is known as an Inverse Square Law. Force weakens like the square of the distance: if you double the distance, the force changes by a factor of 1/(2x2) = 1/4.

4 Orbits and Gravity Gravity is the force which keeps the planets from flying off into space Because the Sun is much more massive then the planets the Sun controls the motion of the planets Gravity always pulls the planet toward the Sun Inertia wants to keep the planet moving in a straight line The balance between gravity and inertia leads to the stable orbit of a planet Throw the ball fast enough and it will go into orbit - The Moon is much closer to the Earth than the Sun. - The inverse square law nature of gravity means that the Moon s orbit is controlled by the mass of the Earth, rather than the Sun. - The force the Moon feels from the Earth is stronger than that it feels from the Sun. [Try and compare the ratio of the forces the Moon feels from the Sun and the Earth to verify this.] The Shapes of Orbits The shape of an object s orbit depends on its velocity perpendicular to the force of gravity A body with a small perpendicular velocity will fall nearly straight in A body with a large perpendicular velocity will overcome the force of gravity and move to a larger distance For closed orbits the shapes will be ellipses Escape Velocity: if the velocity of an object is greater than a certain value, the escape velocity, then gravity is unable to slow down the object enough to prevent it from flying out to deep space. For the Earth the escape velocity is about 11 km/s. Relation of Newton s work to Kepler s 2 nd Law As a planet moves toward the Sun the force of gravity causes it to accelerate along its orbits and it moves faster As a planet moves away from the Sun the force of gravity acts along its orbit and slows it down

5 Relation of Newton s work to Kepler s 3 rd Law Planets with larger average distances from the Sun have longer periods (P 2 =ka 3 ) Since the gravitational acceleration is less they move more slowly along their orbits The orbits are larger Newton s Version of Kepler s 3 rd Law Newton applied his laws of motion and gravity to derive a modified version of Kepler s 3 rd law (which was P 2 = k a 3 ) (Mass1+Mass2)x Period 2 = K(average distance) 3 Here K is a new constant. This equation can also be written as M total P 2 = K a 3 In the solar system the mass of the Sun is so large that M total = M sun +M planet is almost exactly equal to M sun This law allows the determination of masses for distant objects if the orbital properties (P and a) can be measured. Note Kepler s 3rd Law Applies to Any Object Orbiting the Sun (can be on near circular orbits or very elliptical, i.e. very eccentric) P 2 = k a 3, with k = constant, P = period of orbit, and a = average distance of object from Sun Earth has P=1 year, a = 1 AU, so using these units, k=1, and we can write P 2 =a 3 Examples - If a planet has a = 4 AU, then it must have period P = 8 years: 8x8 = 64 = 4x4x4 - This formula applies to any orbit around the Sun, from circular to very eccentric (e.g. comets). Newton s Cosmology Gravity holds the solar system together The Sun is the most massive object so its gravity dominates the solar system The law of Universal Gravitation naturally produces elliptical orbits (Kepler s 1 st law) The law of Universal Gravitation naturally produces Kepler s 2 nd and 3 rd laws Newton thought that beyond the solar system, the universe of stars must be infinite or it would collapse. We shall see later if Newton was right.

6 Complexity to Simplicity For centuries people had tried to understand the unique motions of the planets 1 - They were Gods that had special power over our lives 2 - They were mystical bodies moving in a complex clock work universe with circular orbits, epicycles, and other geometrical devices Not composed of the same material as the Earth Not covered by the same laws of nature as the Earth 3 - They were special objects moving under the control of three laws of planetary motion (Kepler) Complexity to Simplicity 4 - Laws of physics (Motion and Gravitation) described the motion of the planets & much, much more The planets obey the same laws of motion and gravity as any object on the Earth or in the universe The planets are composed of the same types of matter as is the Earth The same laws of motion & gravitation can explain a wide range of phenomena The orbits of planets Tides How to build a bridge or tall building & land people on the Moon [We will see later in the class that our understanding of the physics laws has evolved further since Newton s time due to the work of Albert Einstein.]