>> Hello, class, Professor Anderson here. Let's take a look at an inclined plane problem and let's make it a little more complicated. Let's add a pulley to this incline plane. And we will hang a mass from this pulley, call that M1. And we will have the pulley tied to another mass right there, M2. And let's add some friction on the plane with coefficient Mu K. All right, we don't know exactly which way this thing is going to accelerate, but let's just pick a direction. And, if we get a negative sign when we're all done, then you know you had your picture wrong and it's actually accelerating the other way. So let's say that M2 is big enough that it's going to pull down that way and the entire thing is going to accelerate down the plane. If block M2 is accelerating down the plane, then block M1 is accelerating up with the exact same acceleration A because they're tied together. All right. So, this is what the picture looks like. Let's say that we are trying to figure out what that acceleration A is, in fact, equal to. So how do we do this? Well, we've got our picture. We have our givens, M1, M2, Mu K and theta. The next step is to draw a free body diagram. So let's draw a free body diagram for particle number 1, M1, what are the forces on it? Well, of course, gravity is pulling down and we have tension T in the cable going up. And that's it for particle 1, okay? What about number 2? Number 2, it's on an incline so, again, we want to use our rotated coordinate system. And in that rotated coordinate system, there is going to be a force down the plane and we know that that force is M2G sine of theta. There is another component of gravity which is into the plane and that is M2G cosine of theta. Again, if you can't remember which is which, look at the limits as theta goes to 0 and that should tell you if you have it right or you have it backwards. What other forces are acting on M2? Well, there is the normal force from the plane. There is also tension, T, because it's tied to the cable. And there is also friction, F sub K. And that's all the forces that are acting on M2. And now, with this picture, we can write down Newton's Second Law and see if we can solve for the acceleration A. So let's make a little room here. And we will attack it. All right, let's look at particle number 1. Sum of the forces, it's all in the Y direction for that guy. And it's equal to tension going up minus M1G going down and all that is going to be equal to the mass times the acceleration. And so now you can write the tension T. Tension T, according to this equation, is just M1A plus M1G. All right we like that one and we'll put a box around it. What about number 2. For number 2, this is number 1, number 2, we have sum of the forces in the X direction. Remember we're in this rotated coordinate system. So the X direction means positive X is down to the right. So we have M2G sine theta, okay. And then we have 2 that are opposing that. We have tension T and we also have frictional force FK. And all of that is equal to the mass times its acceleration. We said it's going to accelerate down the plane, okay? That looks pretty good. What about the sum of the forces in the Y direction for that box? We've got N2 in the positive Y. Remember we're in this rotated coordinate system so N2 minus M2G cosine theta and all of that is equal to because it's staying on the incline. It's not jumping up off the incline or falling through the incline. And so now we immediately know what the normal force is on that box. It's M2G cosine theta. All right so now we're getting somewhere. Let's take a look at what is left. Well, it looks like this equation right here is a good place to start. So let's write that up here. M2G sine theta minus T, but we know what T is, it's right there. M1A plus M1G. And now we have to subtract FK, but we know exactly what FK is. Fk is equal to Mu K times N, the normal force which is M2G cosine theta. And all of that is equal to M2A. So now look what we have, we have 1 equation and we can solve this for acceleration A and I'm not going to bore you with all the details, you can double check it on your own. But you do a little bit of math and what you should end up with is the following. A is equal to G times the quantity M2 sine theta minus Mu K M2 cosine theta minus M1. All of that over M1 plus M2. Okay? Double check the math, make sure you get to the same answer. Hopefully that is clear and, if you have any trouble with that one, 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
7. Friction, Inclines, Systems
Systems of Objects on Inclined Planes with Friction
Video duration:
7mPlay a video:
Related Videos
Related Practice