| Title | Author | Created | Published | Tags | | ----------------- | ---------------------------- | ----------------- | ----------------- | ---------------------- | | Energy & Momentum | <ul><li>Jon Marien</li></ul> | February 11, 2025 | February 11, 2025 | [[#classes\|#classes]] | # Energy – Helmets 1 ## Energy and Oscillations  Why does a swinging pendant return to the same point after each swing? The force does work to move the ball\. This increases the ball’s energy\, affecting its motion\.  ## Simple Machines, Work, and Power * A _ simple machine_ multiplies the effect of an applied force\. * For example\, a _lever_ :  A small force applied to one end delivers a large force to the rock\. The small force acting through a large distance moves the rock a small distance\. * A _ simple machine_ multiplies the effect of an applied force\. * For example\, a _pulley_ :  A small tension applied to one end delivers twice as much tension to lift the box\. The small tension acting through a large distance moves the box a small distance\. The _mechanical advantage _ of a simple machine is the ratio of the output force to the input force\.  * For this pulley example\, the mechanical advantage is 2\. * _Work_ is equal to the force applied times the distance moved\. * Work = Force x Distance: <span style="color:#faa368"> _W = F d_ </span> * Work output = Work input <span style="color:#faa368">units: 1 Joule \(J\) = 1 Nm</span>  Only forces _parallel_ to the motion do work\. In this case\, with the block sliding horizontally\, only the 30N part of the diagonal force does work\.  * _Power_ is the rate of doing work * Power = Work divided by Time: <span style="color:#faa368"> _P = W / t_ </span> * <span style="color:#faa368"> </span> <span style="color:#faa368">units: 1 watt \(W\) = 1 J / s</span> **A string is used to pull a wooden block across the floor without accelerating the block. The string makes an angle to the horizontal. Does the force applied via the string do work on the block?**  Yes\, the force __F__ does work\. No\, the force __F__ does no work\. Only part of the force __F__ does work\. You can’t tell from this diagram\. Only the part of the force that is _parallel_ to the distance moved does work on the block\. This is the _horizontal_ part of the force __F__ \. **If there is a frictional force opposing the motion of the block, does this frictional force do work on the block?** Yes\, the frictional force does work\. No\, the frictional force does no work\. Only part of the frictional force does work\. You can’t tell from this diagram\.  Since the frictional force is _antiparallel_ to the distance moved\, it does _negative_ work on the block\. **Does the normal force of the floor pushing upward on the block do any work?** Yes\, the normal force does work\. No\, the normal force does no work\. Only part of the normal force does work\. You can’t tell from this diagram\.  Since the normal force is _perpendicular_ to the distance moved\, it does _no_ work on the block\. **What is the total work done by the 50-N force?** 120 J 160 J 200 J 280 J 0 J  Only the component of force in the direction of motion does work: <span style="color:#ffbf8a">W = F · d </span> <span style="color:#ffbf8a"> = \(40 N\) · \(4 m\) </span> <span style="color:#ffbf8a"> = 160 J</span> \. # Kinetic Energy * _Kinetic energy_ is the energy associated with an object’s motion\. * Doing work on an object increases its kinetic energy\. * Work done = change in kinetic energy  # Potential Energy * If work is done but no kinetic energy is gained\, we say that the _potential energy_ has increased\. * For example\, if a force is applied to lift a crate\, the _gravitational potential energy _ of the crate has increased\. * The work done is equal to the force \(mg\) times the distance lifted \(height\)\. * The gravitational potential energy equals mgh\.  **Work is done on a large crate to tilt the crate so that it is balanced on one edge, rather than sitting squarely on the floor as it was at first. Has the potential energy of the crate increased?**  Yes No Yes\. The center of the crate has been lifted slightly\. If it is released it will fall back and convert the potential energy into kinetic energy\. # Potential Energy  * The term _potential energy_ implies storing energy to use later for other purposes\. * For example\, the _gravitational potential energy _ of the crate can be converted to kinetic energy and used for other purposes\. * An _elastic force_ is a force that results from stretching or compressing an object\. * _Elastic potential energy_ is the energy gained when work is done to stretch a spring\. * The _spring constant\, k\,_ is a number describing the stiffness of the spring\.  * _Conservative forces_ are forces for which the energy can be completely recovered\. * Gravity and elastic forces are conservative\. * Friction is not conservative\.  # Conservation of Energy  * _Conservation of energy_ means the total energy \(the kinetic plus potential energies\) of a system remain constant\. * Energy is conserved if there are no non\-conservative forces doing work on the system\. # A lever is used to lift a rock. Will the work done by the person on the lever be greater than, less than, or equal to the work done by the lever on the rock?  Greater than Less than Equal to Unable to tell from this diagram The work done by the person can never be less than the work done by the lever on the rock\. If there are no dissipative forces they will be equal\. This is a consequence of the conservation of energy\. * Work done in pulling a sled up a hill produces an increase in <span style="color:#ffbf8a"> _potential energy_ </span> of the sled and rider\. * This initial energy is converted to <span style="color:#ffbf8a"> _kinetic energy_ </span> as they slide down the hill\.  * Any work done by frictional forces is <span style="color:#ffbf8a"> _negative_ </span> \. * That work <span style="color:#ffbf8a"> _removes_ </span> mechanical energy from the system\.  # Simple Harmonic Motion # Springs and Simple Harmonic Motion _Simple harmonic motion_ occurs when the energy of a system repeatedly changes from potential energy to kinetic energy and back again\. Energy added by doing work to stretch the spring is transformed back and forth between potential energy and kinetic energy\.  # The horizontal position x of the mass on the spring is plotted against time as the mass moves back and forth. * The _period T _ is the time taken for one complete cycle\. * The _frequency f _ is the number of cycles per unit time\. * The _amplitude _ is the maximum distance from equilibrium\.  # Momentum and Impulse # Collisions and Energy # Collisions How can we describe the change in velocities of colliding football players\, or balls colliding with bats? How does a strong force applied for a very short time affect the motion? Can we apply Newton’s Laws to collisions? What exactly is _momentum_ ? How is it different from force or energy? What does “ _Conservation of Momentum_ ” mean? # What happens when a ball bounces?  When it reaches the floor\, its velocity quickly changes direction\. There must be a strong force exerted on the ball by the floor during the short time they are in contact\. This force provides the upward acceleration necessary to change the direction of the ball’s velocity\. * Forces like this are difficult to analyze: * Strong forces that act for a very short time\. * Forces that may change rapidly during the collision\. * It will help to write Newton’s second law in terms of the total change in velocity over time\, instead of acceleration:  # Momentum and Impulse Multiply both sides of Newton’s second law by the time interval over which the force acts: The left side of the equation is _impulse_ \, the \(average\) force acting on an object multiplied by the time interval over which the force acts\. How a force changes the motion of an object depends on both the size of the force and how long the force acts\. The right side of the equation is the _change in the momentum_ of the object\. The _momentum_ of the object is the mass of the object times its velocity\. <span style="color:#ffebc7">A bowling ball and a tennis ball can have the same momentum\, if the tennis ball with its smaller mass has a much larger velocity\.</span>  # Impulse-Momentum Principle The impulse acting on an object produces a change in momentum of the object that is equal in both magnitude and direction to the impulse\. When a ball bounces back with the same speed\, the momentum changes from <span style="color:#ffbf8a"> __\-mv__ </span> to <span style="color:#ffbf8a"> __mv__ </span> \, so the change in momentum is <span style="color:#ffbf8a"> __2mv__ </span> \.  # Momentum…in space! # Conservation of Momentum  * Does Newton’s third law still hold? * For every action\, there is an equal but opposite reaction\. * The defensive back exerts a force on the fullback\, and the fullback exerts an equal but opposite force on the defensive back\.  * If the net external force acting on a system of objects is zero\, the total momentum of the system is _conserved_ \.  * The impulses on both are equal and opposite\. * The changes in magnitude for each are equal and opposite\. * The total change of the momentum for the two players is zero\. # A 100-kg fullback moving straight downfield collides with a 75-kg defensive back. The defensive back hangs on to the fullback, and the two players move together after the collision. What is the initial momentum of each player?  # What is the initial momentum of each player? Fullback: <span style="color:#ffbf8a">p = mv</span> <span style="color:#ffbf8a"> = \(100 kg\)\(5 m/s\)</span> <span style="color:#ffbf8a"> = 500 kg·m/s</span> <span style="color:#ffbf8a">2</span> Defensive back: <span style="color:#ffbf8a">p = mv</span> <span style="color:#ffbf8a"> = \(75 kg\)\(\-4 m/s\)</span> <span style="color:#ffbf8a"> = \-300 kg·m/s</span> <span style="color:#ffbf8a">2</span>  # What is the total momentum of the system? Total momentum: <span style="color:#ffbf8a">p</span> <span style="color:#ffbf8a">total</span> <span style="color:#ffbf8a"> = p</span> <span style="color:#ffbf8a">fullback</span> <span style="color:#ffbf8a"> \+ p</span> <span style="color:#ffbf8a">defensive back</span> <span style="color:#ffbf8a"> = 500 kg·m/s \- 300 kg·m/s</span> <span style="color:#ffbf8a"> = 200 kg·m/s</span>  # What is the velocity of the two players immediately after the collision?  Total mass: <span style="color:#ffbf8a"> m = 100 kg \+ 75 kg</span> <span style="color:#ffbf8a"> = 175 kg</span> Velocity of both: <span style="color:#ffbf8a"> v = p</span> <span style="color:#ffbf8a">total</span> <span style="color:#ffbf8a"> / m</span> <span style="color:#ffbf8a"> = \(200 kg·m/s\) / 175 kg</span> <span style="color:#ffbf8a"> = 1\.14 m/s</span> # Recoil Why does a shotgun slam against your shoulder when fired\, sometimes painfully? How can a rocket accelerate in empty space when there is nothing there to push against except itself? # Two skaters of different masses prepare to push off against one another. Which one will gain the larger velocity? The more massive one The less massive one They will each have equal but opposite velocities\.  The less massive one\! The net external force acting on the system is zero\, so conservation of momentum applies\. Before the push\-off\, the total initial momentum is zero\. The total momentum after the push\-off should also be zero\. # Recoil is what happens when a brief force between two objects causes the objects to move in opposite directions.  <span style="color:#ffebc7">The lighter object attains the larger velocity to equalize the magnitudes of the momentums of the two objects\.</span> <span style="color:#ffebc7">The total momentum of the system is conserved and does not change\.</span> # Is momentum conserved when shooting a shotgun? <span style="color:#ffebc7">The explosion of the powder causes the shot to move very rapidly forward\.</span> <span style="color:#ffebc7">If the gun is free to move\, it will recoil backward with a momentum equal in magnitude to the momentum of the shot\.</span>  # Gun Recoil <span style="color:#ffebc7">Even though the mass of the shot is small\, its momentum is large due to its large velocity\.</span> <span style="color:#ffebc7">The shotgun recoils with a momentum equal in magnitude to the momentum of the shot\.</span> <span style="color:#ffebc7">The recoil velocity of the shotgun will be smaller than the shot’s velocity because the shotgun has more mass\, but it can still be sizeable\.</span>  # How can you avoid a bruised shoulder? * <span style="color:#ffebc7">If the shotgun is held firmly against your shoulder\, it doesn’t hurt as much\.</span> * <span style="color:#ff0000"> _ WHY?_ </span>  <span style="color:#ffebc7">If you think of the system as just the shotgun and the pellets\, then your shoulder applies a strong external force to the system\.</span> <span style="color:#ffebc7">Since conservation of momentum requires the external force to be zero\, the momentum of this system is not conserved\.</span>  <span style="color:#ffebc7">If you think of the system as including yourself with your shoulder against the shotgun\, then momentum is conserved because all the forces involved are internal to this system \(except possibly friction between your feet and the earth\)\.</span> <span style="color:#ffebc7">With your mass added to the system\, the recoil velocity is smaller\.</span>  # Recoil – Rockets # How does a rocket accelerate in empty space when there is nothing to push against? <span style="color:#ffebc7">The exhaust gases rushing out of the tail of the rocket have both mass and velocity and\, therefore\, momentum\.</span> <span style="color:#ffebc7">The momentum gained by the rocket in the forward direction is equal to the momentum of the exhaust gases in the opposite direction\.</span> <span style="color:#ffebc7">The rocket and the exhaust gases push against each other\.</span> <span style="color:#ffebc7">Newton’s third law applies\.</span>  # Energy – Metal Foam # Elastic and Inelastic Collisions  * <span style="color:#ffebc7">Different kinds of collisions produce different results\.</span> * <span style="color:#ffebc7">Sometimes the objects stick together\.</span> * <span style="color:#ffebc7">Sometimes the objects bounce apart\.</span> * <span style="color:#ffebc7">What is the difference between these types of collisions?</span> * <span style="color:#ffebc7">Is energy conserved as well as momentum?</span> * <span style="color:#ffebc7">A collision in which the objects stick together after collision is called a </span> _perfectly inelastic collision_ <span style="color:#ffebc7">\.</span> * <span style="color:#ffebc7">The objects do not bounce at all\.</span> * <span style="color:#ffebc7">If we know the total momentum before the collision\, we can calculate the final momentum and velocity of the now\-joined objects\.</span> * <span style="color:#ffebc7">For example:</span> * <span style="color:#ffebc7">The football players who stay together after colliding\.</span> * <span style="color:#ffebc7">Coupling railroad cars\.</span>  # Four railroad cars, all with the same mass of 20,000 kg, sit on a track. A fifth car of identical mass approaches them with a velocity of 15 m/s. This car collides and couples with the other four cars. What is the initial momentum of the system? _m_ 5 = 20\,000 kg _v_ 5 = 15 m/s _p_ initial _ _ = _m_ 5 _v_ 5 = \(20\,000 kg\)\(15 m/s\) = <span style="color:#fa4f9f">300\,000 kg·m/s</span> 200\,000 kg·m/s 300\,000 kg·m/s 600\,000 kg·m/s 1\,200\,000 kg·m/s  # What is the velocity of the five coupled cars after the collision? _m_ total = 100\,000 kg _p_ final = _p_ initial _v_ final = _p_ final / _m_ total = \(300\,000 kg·m/s\)/\(100\,000 kg\) = <span style="color:#fa4f9f">3 m/s</span> 1 m/s 3 m/s 5 m/s 10 m/s  # Is the kinetic energy after the railroad cars collide equal to the original kinetic energy of car 5? _KE_ initial = 1/2 _ m_ 5 _ v_ 52 = 1/2 \(20\,000 kg\)\(15 m/s\)2 = <span style="color:#fa4f9f">2250 kJ</span> _KE_ final = 1/2 _ m_ total _ v_ final2 = 1/2 \(100\,000 kg\)\(3 m/s\)2 = <span style="color:#fa4f9f">450 kJ</span> <span style="color:#fa4f9f"> _KE_ </span> <span style="color:#fa4f9f">final</span> <span style="color:#fa4f9f"> ≠ </span> <span style="color:#fa4f9f"> _KE_ </span> <span style="color:#fa4f9f">initial</span> yes no It depends\. No\, in fact it is substantially less than the initial kinetic energy\!  # Elastic and Inelastic Collisions * <span style="color:#ffebc7">Energy is not conserved in a </span> _perfectly inelastic collision_ <span style="color:#ffebc7">\.</span> * <span style="color:#ffebc7">If the objects bounce apart instead of sticking together\, the collision is either </span> _elastic_ <span style="color:#ffebc7"> or </span> _partially inelastic_ <span style="color:#ffebc7">\.</span> * <span style="color:#ffebc7">An </span> _elastic collision_ <span style="color:#ffebc7"> is one in which </span> <span style="color:#ffbf8a"> _no energy is lost_ </span> <span style="color:#ffebc7">\.</span> * <span style="color:#ffebc7">A </span> _partially inelastic collision_ <span style="color:#ffebc7"> is one in which </span> <span style="color:#ffbf8a"> _some energy is lost_ </span> <span style="color:#ffebc7">\, but the objects do not stick together\.</span> * <span style="color:#ffebc7">The </span> <span style="color:#ffbf8a"> _greatest portion of energy is lost_ </span> <span style="color:#ffebc7"> in the </span> _perfectly inelastic collision_ <span style="color:#ffebc7">\, when the objects stick\.</span> * <span style="color:#ffebc7">A ball bouncing off a floor or wall with no decrease in the magnitude of its velocity is an </span> _elastic collision_ <span style="color:#ffebc7">\.</span> * <span style="color:#ffebc7">The kinetic energy does not decrease\.</span> * <span style="color:#ffebc7">No energy has been lost\.</span> * <span style="color:#ffebc7">A ball sticking to the wall is a </span> _perfectly inelastic collision_ <span style="color:#ffebc7">\.</span> * <span style="color:#ffebc7">The velocity of the ball after the collision is zero\.</span> * <span style="color:#ffebc7">Its kinetic energy is then zero\.</span> * <span style="color:#ffebc7">All of the kinetic energy has been lost\.</span> * <span style="color:#ffebc7">Most collisions involve some energy loss\, even if the objects do not stick\, because the collisions are not perfectly elastic\.</span> * <span style="color:#ffebc7">Heat is generated\, the objects may be deformed\, and sound waves are created\.</span> * <span style="color:#ffebc7">These would be </span> _partially inelastic collisions_ <span style="color:#ffebc7">\.</span> # What happens when billiard balls bounce? * <span style="color:#ffebc7">Simplest case: a head\-on collision between the white cue ball and the eleven ball initially at rest\.</span> * <span style="color:#ffebc7">If spin is not a factor\, the cue ball stops and the eleven ball moves forward with a velocity equal to the initial velocity of the cue ball\.</span> * <span style="color:#ffebc7">The eleven ball’s final momentum is equal to the cue ball’s initial momentum\.</span> * <span style="color:#ffebc7">Momentum is conserved\.</span> * <span style="color:#ffebc7">The eleven ball also has </span> * <span style="color:#ffebc7"> a final kinetic energy </span> * <span style="color:#ffebc7"> equal to the cue ball’s </span> * <span style="color:#ffebc7"> initial kinetic energy\.</span> * <span style="color:#ffebc7">Energy is conserved\.</span>  # Collisions at an Angle * Two football players traveling at right angles to one another collide and stick together\. * What will be their direction of motion after the collision?  Add the individual momentum vectors to get the total momentum of the system before the collision\. The final momentum of the two players stuck together is equal to the total initial momentum\. * The total momentum of the two football players prior to the collision is the vector sum of their individual momentums\. * The larger initial momentum has a larger effect on the final direction of motion\.   # Two lumps of clay of equal mass are traveling at right angles with equal speeds as shown, when they collide and stick together. Is it possible that their final velocity vector is in the direction shown?  yes no unable to tell from this graph The final momentum will be in a direction making a 45o degree angle with respect to each of the initial momentum vectors\. # Two cars of equal mass Collide at right angles to one another in an intersection. Their direction of motion after the collision is as shown. Which car had the greater velocity before the collision? Car A Car B Their velocities were equal in magnitude\. It is impossible to tell from this graph\.  Since the angle with respect to the original direction of A is smaller than 45º\, car A must have had a larger momentum and thus was traveling faster\.