>> Hello class, Professor Anderson here. Let's talk about the difference between a horizontal spring and a vertical spring. So, let's say we do the following experiment. Let's say we have a spring with a mass attached to it, and we stretch it out, and we let it go and there's no friction or anything to worry about. And, let's stretch it out to a distance, A. We'll let it go back through equilibrium. We know that it's going to go compress the spring to a distance minus A, okay? So, where is the velocity going to be a maximum? Well, the velocity is a maximum when it goes back through the equilibrium position. And, so if we think about conservation of energy here, what can we say? Ei equals Ef. When we start, we are all spring potential energy one-half KA squared. When we go back through equilibrium we are all kinetic energy; one-half mV max squared. You can quickly solve this for Vmax, right? Multiply by two, divide by the m, we get Vmax equals square root K over m times A. But, we also remember that square root of K over m is our good old angular frequency omega. Now, how does that differ if we take that system and instead we hang it from the roof? So, let's draw the system hanging from the roof. First off, let's just hang the spring, okay? It's hanging at its equilibrium length. And, now let's attach the box. When we attach the box, it stretches, a little bit. It stretches a distance L. And, now let's stretch it by hand a distance A [drawing], okay, from the equilibrium spot down to there. We'll let it go. What's it going to do? Well, we know it's going to oscillate up and down, and in fact, we know it's going to go to a maximum height of A above that equilibrium spot. But, how does the maximum speed compare to our horizontal table? Let's define this equilibrium position as gravitational potential energy there equal to zero, and now let's look at this from a conservation of energy standpoint. So, let's call this lowest position here, position one. We will say that when it comes back up through the equilibrium position that is two, and when it gets up to its maximum height above it that is position three, and let's see what we get for conservation of energy. Before we do that, let's see if we can figure out what this new equilibrium position is. When you hang the mass from the spring there is gravity acting on that mass, but there is the spring force holding it up. And, if it stretches a distance L, then the spring force is equal to kL, and so now look what we have. Those forces have to be zero when it's just hanging there, and so we get kL minus mg equals zero, or kL equals mg. Now, let's do one more little trick which we will use later. Let's multiply both sides of this equation by A. kLA equals mgA, and we'll see why we did that in a second. Okay, so now let's go to our positions one, two, and three and let's use conservation of energy. Okay, so what's the energy at position one? The energy at position one is, well there is no kinetic energy, because it's not moving at the bottom, so that's zero. We do have some spring energy, and how much has that spring stretched out from its original equilibrium? Length, it has stretched out A plus L. And, if we set position two to have zero gravitational potential energy, then we have to subtract mgA from this expression. Okay? And, so that's where we end up with for E one. Alright, what about E two? E two is when the block is moving back upwards. It has kinetic energy at that point. We suspect that's going to be the maximum speed, but we're not totally sure yet. It also has a little bit of spring potential energy, because even when it's at position that spring is still stretched a little bit. It stretched by L, and we have zero potential energy due to gravity, because we said that's where it equals zero. Alright, what about E three? E three is when the block has moved all the way up to the top of its motion, and let's see what we have for those terms. We don't have any kinetic energy. It's come to rest up there, so that one's zero. We might have a little bit of stretch in that spring still, and the amount of stretch is going to be the difference between A and L. So, we have to subtract those two. And, we have some gravitational potential energy, because it's up at a height A, and so we have to add mgA. Okay? And, now we're going to take advantage of this little last statement here, kLA equals mgA. Okay, so let's put in these terms here. So, we have one-half k, A plus L, quantity squared, minus mgA, but mgA we could write as this, kLA. E two, we'll leave that alone for a second. E three becomes what? One-half k, A minus L, quantity squared and then we have plus mgA which is plus kLA. So, now we have all of those energies in terms of the spring constant k, the rest length L, and how far we stretched it, A. And, now let's see if we can solve this thing for velocity. Alright? So, to do that we just set these energies equal, and why don't we take equation one and set it to equation two. So, if we say E one equals E two, what do we have? One-half mV squared plus one-half kLA squared equals E two. E two is right here, and let's write that down here, but let's multiply it out. So, we have one-half kA squared, plus we've got to do the foil on this, so now we have two AL in there, and that two is going to cancel with the one-half out in front. So, we have kAL, and then we have one-half kL squared. And, then we're still subtracting this last term kLA. And, now look what happens. kAL and minus kLA we can get rid of those. And, one-half kL squared is over there as well, so we can get rid of those, and that looks very familiar. That looks like our old equation, and in fact, V is equal to what? Well, we multiply by two, we divide by m, we get V max equals square root of k over m times A which is omega times A which is exactly the same as the horizontal case. So, a vertical spring behaves exactly the same as a horizontal spring, as long as it's all relative to a new equilibrium position, and that's the whole point of this exercise. Alright, hopefully that's clear. If not, come see me in my office. Cheers.
Table of contents
- 0. Math Review31m
- 1. Intro to Physics Units1h 23m
- 2. 1D Motion / Kinematics3h 56m
- Vectors, Scalars, & Displacement13m
- Average Velocity32m
- Intro to Acceleration7m
- Position-Time Graphs & Velocity26m
- Conceptual Problems with Position-Time Graphs22m
- Velocity-Time Graphs & Acceleration5m
- Calculating Displacement from Velocity-Time Graphs15m
- Conceptual Problems with Velocity-Time Graphs10m
- Calculating Change in Velocity from Acceleration-Time Graphs10m
- Graphing Position, Velocity, and Acceleration Graphs11m
- Kinematics Equations37m
- Vertical Motion and Free Fall19m
- Catch/Overtake Problems23m
- 3. Vectors2h 43m
- Review of Vectors vs. Scalars1m
- Introduction to Vectors7m
- Adding Vectors Graphically22m
- Vector Composition & Decomposition11m
- Adding Vectors by Components13m
- Trig Review24m
- Unit Vectors15m
- Introduction to Dot Product (Scalar Product)12m
- Calculating Dot Product Using Components12m
- Intro to Cross Product (Vector Product)23m
- Calculating Cross Product Using Components17m
- 4. 2D Kinematics1h 42m
- 5. Projectile Motion3h 6m
- 6. Intro to Forces (Dynamics)3h 22m
- 7. Friction, Inclines, Systems2h 44m
- 8. Centripetal Forces & Gravitation7h 26m
- Uniform Circular Motion7m
- Period and Frequency in Uniform Circular Motion20m
- Centripetal Forces15m
- Vertical Centripetal Forces10m
- Flat Curves9m
- Banked Curves10m
- Newton's Law of Gravity30m
- Gravitational Forces in 2D25m
- Acceleration Due to Gravity13m
- Satellite Motion: Intro5m
- Satellite Motion: Speed & Period35m
- Geosynchronous Orbits15m
- Overview of Kepler's Laws5m
- Kepler's First Law11m
- Kepler's Third Law16m
- Kepler's Third Law for Elliptical Orbits15m
- Gravitational Potential Energy21m
- Gravitational Potential Energy for Systems of Masses17m
- Escape Velocity21m
- Energy of Circular Orbits23m
- Energy of Elliptical Orbits36m
- Black Holes16m
- Gravitational Force Inside the Earth13m
- Mass Distribution with Calculus45m
- 9. Work & Energy1h 59m
- 10. Conservation of Energy2h 54m
- Intro to Energy Types3m
- Gravitational Potential Energy10m
- Intro to Conservation of Energy32m
- Energy with Non-Conservative Forces20m
- Springs & Elastic Potential Energy19m
- Solving Projectile Motion Using Energy13m
- Motion Along Curved Paths4m
- Rollercoaster Problems13m
- Pendulum Problems13m
- Energy in Connected Objects (Systems)24m
- Force & Potential Energy18m
- 11. Momentum & Impulse3h 40m
- Intro to Momentum11m
- Intro to Impulse14m
- Impulse with Variable Forces12m
- Intro to Conservation of Momentum17m
- Push-Away Problems19m
- Types of Collisions4m
- Completely Inelastic Collisions28m
- Adding Mass to a Moving System8m
- Collisions & Motion (Momentum & Energy)26m
- Ballistic Pendulum14m
- Collisions with Springs13m
- Elastic Collisions24m
- How to Identify the Type of Collision9m
- Intro to Center of Mass15m
- 12. Rotational Kinematics2h 59m
- 13. Rotational Inertia & Energy7h 4m
- More Conservation of Energy Problems54m
- Conservation of Energy in Rolling Motion45m
- Parallel Axis Theorem13m
- Intro to Moment of Inertia28m
- Moment of Inertia via Integration18m
- Moment of Inertia of Systems23m
- Moment of Inertia & Mass Distribution10m
- Intro to Rotational Kinetic Energy16m
- Energy of Rolling Motion18m
- Types of Motion & Energy24m
- Conservation of Energy with Rotation35m
- Torque with Kinematic Equations56m
- Rotational Dynamics with Two Motions50m
- Rotational Dynamics of Rolling Motion27m
- 14. Torque & Rotational Dynamics2h 5m
- 15. Rotational Equilibrium3h 39m
- 16. Angular Momentum3h 6m
- Opening/Closing Arms on Rotating Stool18m
- Conservation of Angular Momentum46m
- Angular Momentum & Newton's Second Law10m
- Intro to Angular Collisions15m
- Jumping Into/Out of Moving Disc23m
- Spinning on String of Variable Length20m
- Angular Collisions with Linear Motion8m
- Intro to Angular Momentum15m
- Angular Momentum of a Point Mass21m
- Angular Momentum of Objects in Linear Motion7m
- 17. Periodic Motion2h 9m
- 18. Waves & Sound3h 40m
- Intro to Waves11m
- Velocity of Transverse Waves21m
- Velocity of Longitudinal Waves11m
- Wave Functions31m
- Phase Constant14m
- Average Power of Waves on Strings10m
- Wave Intensity19m
- Sound Intensity13m
- Wave Interference8m
- Superposition of Wave Functions3m
- Standing Waves30m
- Standing Wave Functions14m
- Standing Sound Waves12m
- Beats8m
- The Doppler Effect7m
- 19. Fluid Mechanics2h 27m
- 20. Heat and Temperature3h 7m
- Temperature16m
- Linear Thermal Expansion14m
- Volume Thermal Expansion14m
- Moles and Avogadro's Number14m
- Specific Heat & Temperature Changes12m
- Latent Heat & Phase Changes16m
- Intro to Calorimetry21m
- Calorimetry with Temperature and Phase Changes15m
- Advanced Calorimetry: Equilibrium Temperature with Phase Changes9m
- Phase Diagrams, Triple Points and Critical Points6m
- Heat Transfer44m
- 21. Kinetic Theory of Ideal Gases1h 50m
- 22. The First Law of Thermodynamics1h 26m
- 23. The Second Law of Thermodynamics3h 11m
- 24. Electric Force & Field; Gauss' Law3h 42m
- 25. Electric Potential1h 51m
- 26. Capacitors & Dielectrics2h 2m
- 27. Resistors & DC Circuits3h 8m
- 28. Magnetic Fields and Forces2h 23m
- 29. Sources of Magnetic Field2h 30m
- Magnetic Field Produced by Moving Charges10m
- Magnetic Field Produced by Straight Currents27m
- Magnetic Force Between Parallel Currents12m
- Magnetic Force Between Two Moving Charges9m
- Magnetic Field Produced by Loops and Solenoids42m
- Toroidal Solenoids aka Toroids12m
- Biot-Savart Law (Calculus)18m
- Ampere's Law (Calculus)17m
- 30. Induction and Inductance3h 37m
- 31. Alternating Current2h 37m
- Alternating Voltages and Currents18m
- RMS Current and Voltage9m
- Phasors20m
- Resistors in AC Circuits9m
- Phasors for Resistors7m
- Capacitors in AC Circuits16m
- Phasors for Capacitors8m
- Inductors in AC Circuits13m
- Phasors for Inductors7m
- Impedance in AC Circuits18m
- Series LRC Circuits11m
- Resonance in Series LRC Circuits10m
- Power in AC Circuits5m
- 32. Electromagnetic Waves2h 14m
- 33. Geometric Optics2h 57m
- 34. Wave Optics1h 15m
- 35. Special Relativity2h 10m
10. Conservation of Energy
Springs & Elastic Potential Energy
Video duration:
10mPlay a video:
Related Videos
Related Practice