Physics 218 Review Prof. Rupak Mahapatra Physics 218, Chapter 3 and 4

Checklist for Final Work out all past finals from webpage Do ALL end of chapter exercises from all chapters The final questions are typically text book style questions Look up your final schedule Physics 218, Chapter 3 and 4

The horizontal and vertical Equations of Motion behave independently Projectile Motion The physics of the universe: The horizontal and vertical Equations of Motion behave independently This is why we use vectors in the first place Physics 218, Chapter 3 and 4

The trick for all these problems is to How to Solve Problems The trick for all these problems is to break them up into the X and Y directions Physics 218, Chapter 3 and 4

Firing up in the air at an angle A ball is fired up in the air with speed Vo and angle Qo. Ignore air friction. The acceleration due to gravity is g pointing down. What is the final velocity here? Physics 218, Chapter 3 and 4

Maximize Range Again Find the minimum initial speed of a champagne cork that travels a horizontal distance of 11 meters. Physics 218, Chapter 3 and 4

Translate: Newton’s Second Law The acceleration is in the SAME direction as the NET FORCE This is a VECTOR equation If I have a force, what is my acceleration? More force → more acceleration More mass → less acceleration Physics 218, Chater 5 & 6

Pulling a box A box with mass m is pulled along a frictionless horizontal surface with a force FP at angle Q as given in the figure. Assume it does not leave the surface. What is the acceleration of the box? What is the normal force? FP Q Physics 218, Chater 5 & 6

2 boxes connected with a string Two boxes with masses m1 and m2 are placed on a frictionless horizontal surface and pulled with a Force FP. Assume the string between doesn’t stretch and is massless. What is the acceleration of the boxes? What is the tension of the strings between the boxes? M2 M1 Physics 218, Chater 5 & 6

The weight of a box A box with mass m is resting on a smooth (frictionless) horizontal table. What is the normal force on the box? Push down on it with a force of FP. Now, what is the normal force? Pull up on it with a force of FP such that it is still sitting on the table. What is the normal force? Pull up on it with a force such that it leaves the table and starts rising. What is the normal force? FP Physics 218, Chater 5 & 6

Atwood Machine Two boxes with masses m1 and m2 are placed around a pulley with m1 >m2 What is the acceleration of the boxes? What is the tension of the strings between the boxes? Ignore the mass of the pulley, rope and any friction. Assume the rope doesn’t stretch. Physics 218, Chater 5 & 6

FFriction = mKineticFNormal Kinetic Friction For kinetic friction, it turns out that the larger the Normal Force the larger the friction. We can write FFriction = mKineticFNormal Here m is a constant Warning: THIS IS NOT A VECTOR EQUATION! Physics 218, Chater 5 & 6

FFriction  mStaticFNormal Static Friction This is more complicated For static friction, the friction force can vary FFriction  mStaticFNormal Example of the refrigerator: If I don’t push, what is the static friction force? What if I push a little? Physics 218, Chater 5 & 6

Two Boxes and a Pulley You hold two boxes, m1 and m2, connected by a rope running over a pulley at rest. The coefficient of kinetic friction between the table and box I is m. You then let go and the mass m2 is so large that the system accelerates Q: What is the magnitude of the acceleration of the system? Ignore the mass of the pulley and rope and any friction associated with the pulley Physics 218, Lecture IX

An Incline, a Pulley and two Boxes In the diagram given, m1 and m2 remain at rest and the angle Q is known. The coefficient of static friction is mu and m1 is known. What is the mass m2? Ignore the mass of the pulley and cord and any friction associated with the pulley m2 m1 Q Physics 218, Lecture IX

Skiing You are the ski designer for the Olympic ski team. Your best skier has mass m. She plans to go down a mountain of angle Q and needs an acceleration a in order to win the race What coefficient of friction, m, do her skis need to have? Q Physics 218, Lecture IX

Is it better to push or pull? You can pull or push a sled with the same force magnitude, FP, but different angles Q, as shown in the figures. Assuming the sled doesn’t leave the ground and has a constant coefficient of friction, m, which is better? FP Physics 218, Chater 5 & 6

Work for Constant Forces The Math: Work can be complicated. Start with a simple case Do it differently than the book For constant forces, the work is: …(more on this later) W=F.d Physics 218, Chapter 7 & 8

2 Dim: Force Parallel to Displacement W = F||d = F.d = Fdcosq where q is the angle between the net Force and the net displacement. You can think of this as the force component in the direction of the displacement. Force Force Rotate Displacement F|| = Fcosq Displacement Physics 218, Chapter 7 & 8

Find the work: Calculus To find the total work, we must sum up all the little pieces of work (i.e., F.d). If the force is continually changing, then we have to take smaller and smaller lengths to add. In the limit, this sum becomes an integral. Total sum  Integral Physics 218, Chapter 7 & 8

Non-Constant Force: Springs Springs are a good example of the types of problems we come back to over and over again! Hooke’s Law Force is NOT CONSTANT over a distance Some constant Displacement Physics 218, Chapter 7 & 8

Work done to stretch a Spring How much work do you do to stretch a spring (spring constant k), at constant velocity (pulled slowly), from x=0 to x=D? D Physics 218, Chapter 7 & 8

Work Energy Relationship If net positive work is done on a stationary box it speeds up. It now has energy Work Equation naturally leads to derivation of kinetic energy Kinetic Energy = ½mV2 Physics 218, Chapter 7 & 8

Work-Energy Relationship If net work has been done on an object, then it has a change in its kinetic energy (usually this means that the speed changes) Equivalent statement: If there is a change in kinetic energy then there has been net work on an object Can use the change in energy to calculate the work Physics 218, Chapter 7 & 8

W= DKE Kinetic Energy = ½mV2 Summary of equations Kinetic Energy = ½mV2 W= DKE Can use change in speed to calculate the work, or the work to calculate the speed Physics 218, Chapter 7 & 8

Conservation of Mechanical Energy For some types of problems, Mechanical Energy is conserved (more on this next week) E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick K2+U2 = K1+U1 Conservation of Mechanical Energy E2=E1 Physics 218, Chapter 7 & 8

Problem Solving E = K + U = ½mv2 + mgy What are the types of examples we’ll encounter? Gravity Things falling Springs Converting their potential energy into kinetic energy and back again E = K + U = ½mv2 + mgy Physics 218, Chapter 7 & 8

BEFORE and AFTER diagrams Problem Solving For Conservation of Energy problems: BEFORE and AFTER diagrams Physics 218, Chapter 7 & 8

Potential Energy A brick held 6 feet in the air has potential energy Subtlety: Gravitational potential energy is relative to somewhere! Example: What is the potential energy of a book 6 feet above a 4 foot high table? 10 feet above the floor? DU = U2-U1 = Wext = mg (h2-h1) Write U = mgh U=mgh + Const Only change in potential energy is really meaningful Physics 218, Chapter 7 & 8

Other Potential Energies: Springs Last week we calculated that it took ½kx2 of work to compress a spring by a distance x How much potential energy does it now how have? U(x) = ½kx2 Physics 218, Chapter 7 & 8

EHeat+Light+Sound.. = -WNC Energy Summary If work is done by a non-conservative force it does negative work (slows something down), and we get heat, light, sound etc. EHeat+Light+Sound.. = -WNC If work is done by a non-conservative force, take this into account in the total energy. (Friction causes mechanical energy to be lost) K1+U1 = K2+U2+EHeat… K1+U1 = K2+U2-WNC Physics 218, Lecture XII

Force and Potential Energy If we know the potential energy, U, we can find the force This makes sense… For example, the force of gravity points down, but the potential increases as you go up Physics 218, Lecture XIII

Mechanical Energy We define the total mechanical energy in a system to be the kinetic energy plus the potential energy Define E≡K+U Physics 218, Lecture XIII

Conservation of Mechanical Energy For some types of problems, Mechanical Energy is conserved (more on this next week) E.g. Mechanical energy before you drop a brick is equal to the mechanical energy after you drop the brick K2+U2 = K1+U1 Conservation of Mechanical Energy E2=E1 Physics 218, Lecture XIII

Friction and Springs A block of mass m is traveling on a rough surface. It reaches a spring (spring constant k) with speed Vo and compresses it a total distance D. Determine m Physics 218, Lecture XV

Robot Arm A robot arm has a funny Force equation in 1-dimension where F0 and X0 are constants. The robot picks up a block at X=0 (at rest) and throws it, releasing it at X=X0. What is the speed of the block? Physics 218, Lecture XV

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Overview: Rotational Motion Take our results from “linear” physics and do the same for “angular” physics Analogue of Position ← Velocity ← Acceleration ← Force Mass Momentum Energy Start here! Chapters 1-3 Physics 218, Chapter 12

Velocity and Acceleration Physics 218, Chapter 12

Right-Hand Rule Right-hand Rule Yes! Define the direction to point along the axis of rotation Right-hand Rule This is true for Q, w and a Physics 218, Chapter 12

Uniform Angular Acceleration Derive the angular equations of motion for constant angular acceleration Physics 218, Chapter 12

Rolling without Slipping In reality, car tires both rotate and translate They are a good example of something which rolls (translates, moves forward, rotates) without slipping Is there friction? What kind? Physics 218, Chapter 12

Derivation The trick is to pick your reference frame correctly! Think of the wheel as sitting still and the ground moving past it with speed V. Velocity of ground (in bike frame) = -wR => Velocity of bike (in ground frame) = wR Physics 218, Chapter 12

Centripetal Acceleration “Center Seeking” Acceleration vector= V2/R towards the center Acceleration is perpendicular to the velocity R Physics 218, Chapter 12

Circular Motion: Get the speed! Speed = distance/time  Distance in 1 revolution divided by the time it takes to go around once Speed = 2pr/T Note: The time to go around once is known as the Period, or T Physics 218, Chapter 12

w radians/sec  f = w /2p More definitions Frequency = Revolutions/sec Period = 1/freq = 1/f Physics 218, Chapter 12

Ball on a String A ball at the end of a string is revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t. What is the centripetal acceleration? Physics 218, Chapter 12

The Trick To Solving Problems Physics 218, Chapter 12

Banking Angle You are a driver on the NASCAR circuit. Your car has m and is traveling with a speed V around a curve with Radius R What angle, Q, should the road be banked so that no friction is required? Physics 218, Chapter 12

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4

Physics 218, Chapter 3 and 4