Chapter 3 Energy and Conservation Laws
Conservation laws • The most fundamental ideas we have in physics are conservation laws. • Statements telling us that some quantity does not change. • Conservation of mass states: • The total mass of an isolated system is constant. • To apply these, we must define a “system.”
Conservation laws, cont’d • A system is just a collection of objects we decide to treat at one time. • The tanker and fighter can represent a system. • The fuel leaving the tanker goes into the fighter:mass is conserved.
Linear momentum • Linear momentum is defined as the product of an object’s mass and its velocity. • We typically just say momentum.
Linear momentum, cont’d • Momentum is a measure of an object’s state of motion. • Consider an object whose momentum is 1 kg·m/s • This could be a 0.005 kg bullet traveling at 200 m/s. • This could be a 0.06 kg tennis ball traveling at 16.7 m/s.
Linear momentum, cont’d • Newton’s 2nd law is closely related to momentum. • The net external force acting on an object equals the rate of change of linear momentum:
Linear momentum, cont’d • How is this related to F = ma? • So, F = ma holds only if the object’s mass remains constant. • Not a rocket. • Not really a car but its close enough.
ExampleExample 3.1 Let’s estimate the average force on a tennis ball as it is served. The ball’s mass is 0.06 kg and it leaves the racquet with a speed of 40 m/s. High-speed photography indicates that the contact time is about 5 milliseconds.
ExampleExample 3.1 ANSWER: The problem gives us: The force is:
Linear momentum, cont’d • This tells why we must exert a force to stop an object or get it to move. • To stop a moving object, we have to bring its momentum to zero. • To start moving an object, we have to impart some momentum to it.
Linear momentum, cont’d • It also tells us that we can change the momentum using various forces and time intervals: • Use a large force for a short time, or • Use a small force for a long time.
Conservation of linear momentum • The Law of Conservation of Linear Momentum states: The total linear momentum of an isolated system is constant. • Isolated implies no external force:
Conservation of linear momentum, cont’d • This law helps us deal with collisions. • If the system’s momentum can not change, the momentum before the collision must equal that after the collision.
Conservation of linear momentum, cont’d • We can write this as: • To study a collision: • Add the momenta of the objects before the collision. • Add the momenta after the collision. • the two sums must be equal.
ExampleExample 3.2 A 1,000 kg car (car 1) runs into the rear of a stopped car (car 2) that has a mass of 1,500 kg. Immediately after the collision, the cars are hooked together and have a speed of 4 m/s. What was the speed of car 1 just before the collision?
ExampleExample 3.2 ANSWER: The problem gives us: The momentum before: The momentum after:
ExampleExample 3.2 ANSWER: Conserving momentum
ExampleExample 3.2 DISCUSSION: Both cars together have more mass than just car 1. Since both move away at 4 m/s, the lighter car 1 must have a greater speed before the collision.
Conservation of linear momentum, cont’d • How do rockets work? • The exhaust exits the rocket at high speed. • We need high speed because the gas has little mass. • The rocket moves in the opposite direction. • Not as fast as thegas b/c more mass.
Work • Imagine using a lever to lift a heavy object. • The lever allows us to exert less force than the object actually weighs. • This sounds like “free money.”
Work, cont’d • There’s a catch: • We have to apply our force through a greater distance than the rock moves. • So there must be some connection between force and distance.
Work, cont’d • The force multiplied by the distance moved is the same for both:
Work, cont’d • We have the same situation for placing a barrels on a loading dock:
Work, cont’d • Work is defined as the product of force and the distance through which the force moves an object in the direction of the force.
Work, cont’d • The units of work: • Metric • SI: joule (J = N·m), • erg (= 10-7 J), • calorie (cal = 4.186 J), • kilowatt-hour (kWh = 0.278 J). • English: • foot-pound (ft·lb), • British thermal unit (Btu).
Work, cont’d • From the definition of work:
ExampleExample 3.3 Because of friction, a constant force of 100 newtons is needed to slide a box across a room. If the box moves 3 meters, how much must be done?
ExampleExample 3.3 ANSWER: The problem gives us: The required work is:
Work, cont’d • Recall that force is a vector. • Involves magnitude and direction. • Work is just that part of the force in the direction of the displacement. • Work is not a vector — it’s a scalar. • But the sign of the work does depend on the relative directions.
Work, cont’d • If the force and distance are in the same direction, the force does positive work. • If the force and distance are in the opposite direction, the force does negative work.
Work, cont’d • If the force is not in the direction of the direction, the force does no work. • The string’s tension is toward the center of the circle. • The ball moves along the circle’s circumference. • So, the tension does no work.
Work, cont’d • You do positive work (in the physics-sense) when you lift the create. • You do NO work (in the physics-sense) when you carry the crate. • You do negative work when you set the crate down.
Work, cont’d • When you throw or catch a ball, you do work on the ball. • Your hand exerts a force on the ball. • You exert that force through the throwing or catching distance. • If you’re strong, you don’t need the same distance because of the larger force.
ExampleExample 3.4 Let’s say that the barrel has a mass of 30 kg and that the height of the dock is 1.2 meters. How much work would you do when lifting the barrel?
ExampleExample 3.4 ANSWER: The problem gives us: The required work is:
ExampleExample 3.4 DISCUSSION: You would do the same amount of work rolling the barrel up the ramp. You would only have to exert a force of 150 N instead of the entire 300 N. But you have to exert that smaller force over a distance of 2.4 m.
ExampleExample 3.5 In Example 2.2 we used Newton’s 2nd law to compute the force needed to accelerate a 1,000-kg car from 0 to 27 m/s in 10 seconds. The answer was 2,700 N. How much work is done?
ExampleExample 3.5 ANSWER: The problem gives us: To find work we use: But we need the distance the car moved.
ExampleExample 3.5 ANSWER: Recall that The work is
ExampleExample 3.5 DISCUSSION: In reality, this is smaller than the energy the engine must generate. The engine must overcome its internal friction — a loss of energy. Most cars are about 30% efficient. So you need (364 kJ)/(0.3) = 1.2 MJ to actually accelerate this car.
Energy • Energy is defined as the measure of a system’s ability to do work. • We use the symbol E to represent energy. • Energy has the same units as work: • Joule for SI, ft·lb for English.
Energy, cont’d • There are various types of energy. • Kinetic energy is the energy associated with an object’s motion. • We use the symbol KE. • Potential energy is energy associated with the system’s position or orientation. • We use the symbol PE.
Kinetic energy • The formula for kinetic energy is: • m is the object’s mass. • v is the object’s speed.
ExampleExample 3.6 In Example 3.5 we computed the work that is done on a 1,000-kg car as it accelerates from 0 to 27 m/s. Find the car’s kinetic energy when it is traveling at 27 m/s.
ExampleExample 3.6 ANSWER: The problem gives us: The kinetic energy is:
ExampleExample 3.5 DISCUSSION: This equals the (ideal) work required to get the car up to speed. We could determine how much work is required by finding the kinetic energy of the car. This is the idea of energy conservation.
Gravitational potential energy • Gravitational potential energy equals the work done by the gravity. • If you lift an object, you must apply a force at least equal to the object’s weight: • Lifting it through a distance d, the work is
Gravitational potential energy, cont’d • Note that we only deal with the distance through which the object moves. • The brick has 14.7 J of PE relative to the table top. • It has 44.1 J of PE relative to the floor.
Gravitational potential energy, cont’d • Where you say an object has zero PE is arbitrary. • We only care about the change. • Let’s say the ground is at zero PE. • In the hole, the ball has negative PE. • It is below the reference level.
ExampleExample 3.7 A 3-kg brick is lifted to a height of 0.5 meters above a table that is 1.0-m tall. Find the gravitational potential energy relative to the table and the floor.