Chapter 5: The Law of Gravity & Momentum PDF

Chapter 5: The Law of Gravity & Momentum Physics 100 Lecture Note 1  Why do things fall: The idea of gravity ▪ Aristotle: it the objects’ of nature to seek out the earth’s center in the way that a thirsty person seeks out water. ▪ Law of inertia: In the absence of externa...

Chapter 5: The Law of Gravity & Momentum Physics 100 Lecture Note 1  Why do things fall: The idea of gravity ▪ Aristotle: it the objects’ of nature to seek out the earth’s center in the way that a thirsty person seeks out water. ▪ Law of inertia: In the absence of external influence, an object at rest will stay at rest and an object moving at a constant velocity (constant speed and direction) will continue without change. ▪ Observation: A falling object does not obey the law of inertia ▪ Consequence: A falling object must be subject to an external force Such force changes only the speed of the object (speed increases) Such force does not change the direction of the falling object. ▪ Conclusion: Objects fall because of a force pulling them downwards ▪ This force called the force of gravity ▪ Next, we will try to understand the nature of this force Physics 100 Lecture Note 2  Falling versus orbiting ▪ A stone released from any altitude falls to the ground ▪ An artificial satellites that go around the earth do fall ▪ If both are under the influence of the pull of gravity. Why does one fall and the other does not? ▪ Are the two types of motion related? Physics 100 Lecture Note 3  Orbital motion of the moon ▪ The force of gravity of the earth pulls the moon towards the earth ▪ But the moon has a sideways speed of 1.022 km/s (or 3679.2 km/h) ▪ Moon’s average distance from Earth's center is about 385000km ▪ Without the force of gravity, the moon would fall straight towards the earth ▪ Because of its high speed all that the force of gravity does to its motion is to bend it Physics 100 Lecture Note 4  Question A force of gravity between the sun and its planets holds the planets in orbit around the sun. If that force of gravity suddenly disappeared, in what kind of path would the planets move? Each planet would move in a straight line at constant speed. 12 Physics 100 Lecture Note 5 Physics 100 Lecture Note 6 Physics 100 Lecture Note 7  The law of gravity Statement of the Law ▪ Between any two objects there is an attractive force that is proportional to the product of the two objects’ masses and is inversely proportional to the square of the distance between them Physics 100 Lecture Note 8  Earth’s gravity changes with altitude Gravity vs altitude Physics 100 Lecture Note 9 Example: ▪ The acceleration due to gravity of an object with mass m on the surface of the earth can be calculated using Newton’s law 𝐹 𝑎= 𝑚 Where 𝑚𝑀 𝐹= 𝐺 2 𝑟 ▪ The mass of the earth is 𝑀 = 5.98 × 1024 𝑘𝑔 ▪ The distance between an object on the surface of the earth and the earth is 𝑟 = 6.37 × 106 ▪ We can calculate the acceleration of gravity by substituting these values into Newton’s law and the law of gravity Physics 100 Lecture Note 10 Example Cont..: 𝑀 6.672 × 10−11 ∗ 5.98 × 1024 𝑎=𝐺 2= 𝑟 6.37 × 106 2 = 9.8 𝑚/𝑠 2 ▪ The acceleration due to gravity of an object does not depend on the mass of the object ▪ Remember the feather and the bowling ball falling at the same rate when air is removed Physics 100 Lecture Note 11  Weight ▪ What is the force of gravity due to my weight? Let calculate it. ▪ Let’s assume your mass is 80kg ▪ Your weight is the force of gravity the earth is pulling you with. 𝑚𝑀 𝐹=𝐺 2 𝑟 6.672 × 10−11 × 80 × 5.98 × 1024 = 6.4 × 106 2 = 784𝑁 Physics 100 Lecture Note 12  Force of gravity between two people ▪ What is the gravitational force between you and your neighbor? ▪ Let’s calculate it. ▪ Assume both you and your neighbor have the same mass of 60kg ▪ If you are one meter apart, the force you are pulling each other with is 𝑚𝑀 𝐹=𝐺 2 𝑟 6.672 × 10−11 × 60 × 60 = 1 2 = 0.00000024𝑁 ▪ No wonder that we don’t observe the force of gravity between ordinary objects Physics 100 Lecture Note 13  Momentum Momentum refers to the quantity of motion that an object has. A sports team that is on the move has the momentum. If an object is in motion (on the move) then it has momentum. The motion of a large truck moving at 40 km/hr is different from the motion of a small car moving next to it in the same highway at the same speed. What is different? The velocities are the same (speeds are the same and the directions of motion are the same), so their motions are similar in this respect. However, if we try to slow them down or speed them up to a different velocity, we need to use different forces because they have different masses. So in this respect the motions of the truck and the small car are different. We define the quantity “momentum” in terms of the velocity and the mass as: 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑚𝑎𝑠𝑠 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 Or in symbols 2 p = mv Physics 100 Lecture Note 14  Momentum Momentum ( p ) : Is defined as the product of mass and velocity It is a vector quantity with the same direction as the velocity of the object The units are kg m s–1 Momentum = Mass × Velocity p = mv Physics 100 Lecture Note 15  Example Momentum = Mass × Velocity p = (1000 kg)(16 m/s) p = 16,000 kg m/s Physics 100 Lecture Note 16  Equivalent Momentum Car: m = 1800 kg; v = 80 m/s p = 1.44 ·105 kg · m/s Bus: m = 9000 kg; v = 16 m/s p = 1.44 ·105 kg · m /s p = mv Train: m = 3.6·104 kg; v = 4 m/s p = 1.44 ·105 kg · m/s Physics 100 Lecture Note 17  Conservation of momentum The concept of momentum is one of the most useful concepts in physics. It is importance derives from the fact that when object collide, they can exchange momentum, but the total amount of momentum will stay the same. 5 Physics 100 Lecture Note 18  Types of Collisions 1- Elastic collision _A “No stick” type collision Spbefore = Spafter m1vo1 + m2 vo 2 = m1v1 + m2 v2 (1000)( 20) + 0 = (1000)(v1 ) + (3000)(10) − 10000 = 1000v1 v1 = -10 m/s Physics 100 Lecture Note 19  Types of Collisions 2- Inelastic Collision- A “stick together ” type of collision What is the final speed for the car and truck ? Physics 100 Lecture Note 20 Inelastic Collision- A “stick together ” type of collision Spbefore = Spafter m1vo1 + m2 vo 2 = (m1 + m2 )total vT (1000)( 20) + 0 = ( 4000)vT 20000 = 4000vT vT = 5 m/s Physics 100 Lecture Note 21