Lecture 12 Friction &Circular Exam Remarks Units, units, units Dynamics Drawing graphs with arrows and labels FBDs: ma is not a force! r(r) given, derive v(t), a(t) Circular motion: the velocity vector is NOT constant
Dynamics with Friction
Contact bubbles : how many contacts are there with the outside world
How big are the forces here? W W/2 W/3 Forces are different
How big is the green force? W 2W W/3 None of the above
What is the difference between static and kinetic friction? m: Static friction is stationary at its point of refrence, kinetic friction is friction against something that is moving against a surface. kb: static friction is the frictional force that acts on an object at rest and keeps it at rest. static friction is the frictional force that acts on an object in motion and works to slow that object down.
Friction Friction is proportional to normal force Independent of surface area to a good approx. Direction: antiparallel to direction of (attempted) motion Two types: Static friction: object resists attempts to get it moving Kinetic friction: resists motion by constituting a constant decelerating force
Example: The inclined plane with friction Consider a block of 5 kg on an incline (20 degrees) starting from rest 2.5m upwards along the incline from its basis. The coefficient of kinetic friction is 0.2. Q: How long does it take the block to slide down the incline? This is a dynamics problem: given is the physical situation, i.e. (implicitly) the forces acting on the object of interest
Uniform Circular Motion v α r α
Lecture 13 Circular Dynamics & Work Lab Manual: Rotational Dynamics
Dynamics of circular motion Since a = v 2 /r, we can write down the force But: we don t know what it is, i.e. what entity provides it Centripetal Force has constant strength, but changing direction Examples Car turning corner Space shuttle orbiting Earth
Example: Simulating gravity in Need a = 1g Have r =10m spaceship with r=10m a=4π 2 r /T 2 => T= 2 π r/g = 2 π sec Spaceship needs to turn roughly 10 a minute
Example: Banking a curve If a car on a road goes around a curve (segment of circle), static friction has to supply the centripetal force If the curve is at an angle ( banked ), the normal force can supply the centripetal force (for a given speed and radius)
What is work? Gigi: Work is done by a constant force acting in the same direction as the displacement. It is the work done by a constant force as the product of the force magnitude and the displacement. LI: "Work" is a measure of energy transfer when an object is displaced by an external force. Work is the product of the force magnitude and the displacement magnitude, or the change in the object's kinetic energy.
Work Work W is energy transferred to or form an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work.
Work Work is accomplished by a force applied over some distance: see Man pulling Fire Engine Work is scalar product between force vector and displacement vector : W = F d Angle matters! Work is a number or scalar (with units Joules=J): it can be zero, even negative! Maximal work if force is parallel to displacement
What is kinetic energy? K: Kinetic energy is known as the energy of motion. When an object is moving vertically or horizontally the object has kinetic energy. It is also a scaler product and has no direction but is decided by the magnitude alone. 234567: kinetic energy (K=(1/2)(mv)^2) is dependant on the mass and speed on an object, as seen in the equation.
Kinetic Energy K= ½ m v 2 Units: kg m 2 /s 2 = J E.g: Baseball of 100g at 20 m/s: K=0.5(0.1kg)(400 m 2 /s 2 ) = 20 J
Work & Kinetic Energy Object in motion (relative to another) can do work on contact train on train in crash, etc. Quantitative definition: W= Fd = (ma)d ; use v f2 = v i2 +2ad = m{(v f2 v i2 )/2d}d = ½ mv f2 ½ m v i 2 = K f - K i
Two marbles, one twice as heavy as the other, are dropped to the ground from the top of a building. Just before hitting the ground, the heavier marble has as much kinetic energy as the lighter one. twice as much kinetic energy as the lighter one. half as much kinetic energy as the lighter one. four times as much kinetic energy as the lighter one.
Double index notation and net work There is a force involved, so there are two indices Need to keep it straight: who is doing the work ON what? Still 2 indices necessary and same rule: swap indices, obtain minus sign! Also: net force results in net work Lots of work can be done on system, but might add up to zero
Work done by person on spring vs Work done by spring on person F= kx is Hooke s law It is the force of a spring, i.e. exerted by the spring on something else: F p,s = kx = F s,p W p,s = W s,p
What is the "work-energy theorem"? TP: That the net work on an object causes the change in kinetic energy of that object KP: it is a theorem on change of kinetic energy use to find work
Lecture 14 Kinetic Energy & Work Work and KE in straight-line motion Work and KE with varying forces Work done along a curved path
Example Group Work Calculate the kinetic energy of a baseball (100g) having velocity components v x = 3 m/s, v y = 4 m/s. K= ½ m v 2 Need to convert to kg to get SI unit Joules (J) Sign of velocity does not matter! In higher dimensions v 2 = v x2 +v y2 +v 2 z
Work-energy theorem Net work done on an object is equal to the change of its kinetic energy Note that work and hence change of KE can be zero or even negative!
Interpretation of Kinetic Energy (KE) KE (K=½ m v 2 ) of a particle is the total work that was done to accelerate the particle from rest to its present speed v. Also, the kinetic energy of a particle is equal to the amount of work that a particle can do in the process of being brought to rest (see: hammer driving a nail into wood)
Work done by a varying force is some of small amounts done over small displacements
Varied Force
Example: Work done by Force Work is area under curve (sometimes neg.!) Calculate (area of triangles and rectangles): height in N, width in m Or: count boxes: 4 neg boxes (J) 12 pos boxes (J) So 8 Joules total (Here assume force and motion are in x direction)
Work done by spring F ps d on person F p,s = k x (Force on person by spring, but need to apply F s,p = k x to stretch spring by amount x) W p,s = F p,s dx = W s = kx dx Don t confuse integration variable with value of variable: W s = xi xf kx dx = ½ k x i 2 ½ k x f 2 If x i = 0 then W s = ½ k x 2 Also consider graph: straight line, area of triangle
Example Consider a mass of 2kg on a spring of stiffness 100 N/m on a frictionless table moving in x-direction according to Hooke s law. What is the work the spring is doing on the mass as it id moved 10cm away from the equilibrium point? (A B) Answer: W s = ½ k x 2 = ½ (100N/m)( 0.1m) 2 = 0.5 Nm = 0.5 J
Cross-Check Consider a mass of 2kg on a spring of stiffness 100 N/m on a frictionless table moving in x-direction according to Hooke s law. What is the work the spring is doing on the mass as it is moved 10cm away from the equilibrium point? Answer: W s = ½ k x 2 = 0.5 J Negative?! Does this seem reasonable? No, because it means that the kinetic energy becomes smaller, i.e. negative! No, because there is a force on the mass accelerating it in x direction Yes, because (fill in your reason)
Example: Resolution Consider a mass of 2kg on a spring of stiffness 100 N/m on a frictionless table moving in x-direction according to Hooke s law. What is the work a person has to do on the mass to move it 10cm away from the equilibrium point? Answer: W = ½ k x 2 = 0.5 J This one is positive! Both spring and person are exerting forces on the mass resulting in zero net work, the mass moves and is stationary before and after Work-Energy theorem saved: W=ΔK=0.
Work done along a path
Power The rate at which work is done, i.e. energy is transferred P = dw/dt = (F dx)/dt = F v [P] = J/s =W (Watts) Also: P =de/dt Change in energy per unit time Energy change means energy is being transformed from one form to another or transferred from one object to another
Examples How much work can a horse do in one hour? 1 hp = 746W = 746 J/s Work done in one hour = 746 W*1h= 746 J/s *3600s=2686 kj At which rate is (electric) energy transformed into thermal energy in a 100W light bulb? 100 J are transformed per second How much of the energy goes into EM waves aka photons aka light? Only 5%!
Lecture 15 Potential Energy
On your electric bill it states how many kilo Watt hours (kwh) you used. That means you pay for The electric power delivered to your house The electric energy delivered to your house The electric force delivered to your house None of the above
Lecture 16 Potential Energy
Group Work: Work-energy theorem worksheet
Lecture 17 Energy Conservation
A ball is pushed up an incline from rest to a higher position where it resides. What work does gravity on Positive Zero Negative Depends on angle the ball?
System and potential Energy A System consists of more than one particle Typically a particle-like object and the rest, like bowling ball and Earth, or electron in electric field of a capacitor Force acts between particle and rest of system When the system configuration changes, the force does work W 1 on the particle thereby transferring energy When the configuration change is reversed, the force reverses the energy transfer and does work W 2
Potential Energy Work done around any closed path is zero for conservative forces For conservative forces a function exists that describes the amount of energy stored in a certain configuration (sometimes called a system) involving these forces We can calculate how much work it took to configure the configuration Analogy: building a building costs money because it takes work to build it.
Two types of forces The definition of potential energy does not make sense for all forces When the system configuration changes, the force does work W 1 on the particle transferring energy When the configuration change is reversed, the force reverses the energy transfer and does work W 2 Classify forces by requiring W 1 = W 2 Forces that meet the req.: conservative forces All others: non-conservative forces
Which of the forces below is not conservative? Gravity Friction Force of Spring (Hooke s law) All are conservative
If a particle moves from point A to point B and back via a different path to point A, the work done by a conservative force on the particle is Zero Negative Positive Depends on path taken
Subtle difference Work Kinetic energy theorem ΔK = W net Definition of potential energy (change) ΔU = W by system
A ball is pushed up an incline from rest to a higher position where it resides. What is the net work done on the ball? Positive Zero Negative Depends on angle
Gravitational potential energy When lifting up a stone we do work against gravity: W=Fd=mgh This is energy transferred into the system and stored into the configuration stone sits at height h above the Earth (Stone-Earth system)
Example: Ball thrown vertically System: ball and Earth Particle: the ball Force between particle and rest of the system: gravitational force Work done by force on rising ball: negative Work done by force on falling ball: positive Change in pot. Energy of rising ball: positive Change in pot. Energy of falling ball: negative
Conclusion: Mechanical energy E=K+U is conserved here! System: ball and Earth Particle: the ball Force between particle and rest of the system: gravitational force Work done by force on rising ball: negative Work done by force on falling ball: positive Change in pot. Energy of rising ball: positive Change in pot. Energy of falling ball: negative Note: no external forces, so W net =W by system so E=K+U=const, since ΔK+ΔU =W net W by system = 0
Is Mechanical Energy always conserved when no external forces are doing work? no external forces, so W net =W by system so E=K+U=const, since ΔK+ΔU =W net W by system = 0 Looks like a math proof, but loophole: there are forces that do not have a description in terms of potential energy, yet they do work! Non-conservative forces, e.g. friction
Mechanical Energy is conserved, E mech = K+U if! Necessary & sufficient conditions: Only conservative forces cause energy transfers System is isolated from environment, i.e. not external forces cause energy changes inside system Then can use to find ΔU= W and ΔK= W K i +U i = K f +U f
Potential energy of a spring Compressing a spring we do work against the elastic forces in the wire This is energy transferred into the system and stored into the configuration spring is compressed by a distance x ΔU= W spring = ( k x) dx = ½ k x 2 f - ½ k x 2 i Often use equilibrium point as reference, then U 0 = ½ k x 2 0 U= ½ k x 2 Now x is the distance by which spring is stretched/compressed
Examples: Spring & Block and Pendulum Block & Spring Simulation Energy continuously transferred U K Pendulum Simulation
Energy Bar Charts Intro Example: Mass on a spring 3 forms of energy: K, U grav, U spring
Reference point for Potential Energy Only differences of potential energy are relevant can define zero where we want Typical examples: U=0 initially U=0 at the final position U=0 at a special point
Lecture 18: Solving Problems using Conservation of Energy
Energy Bar Charts Fill in the Energy Bar chart for the following : Ball falling from height h=h to height h=g, define U to be zero at floor (h=0), i.e. U(0)=0 Ball falling from height h=h to height h=g, define U to be zero at initial height (H), U(H)=0 Ball falling from height h=h unto spring of length L compressing it to length d, U(0)=0
Now translate energy bar charts into equations and solve problems!
Energy Problem Solving 1. Draw a picture 2. Determine the system: objects and forces on them 3. What is the unknown? 4. Choose initial and final positions 5. Choose convenient reference frame for potential energy 6. Draw an energy bar chart 7. If mech. Energy is conserved: K+U=K+U 8. If not: K+U+W by nc forces =K f +U f 9. Solve for the unknown
Demo: Track with same initial & final positions, but different paths Which ball takes less time? Ball on straight track Ball on dipped track Both take the same time
Potential Energy & Force Since ΔU = -W = - F Δx we have F = - du/dx in the limit of small Δx Example: U = mgy F = - mg This means U(x) and F(x) contain the same information! (Cf. x(t) and v(t)) In 3D: U is a number, F a vector take 3D derivative vector calculus
Reading Potential Energy Graphs In the simulation A Particle Moving Along an X Axis for the particle with medium energy, which is a true statement? a. The particle has maximal kinetic energy at x 3 b. The sum of the kinetic and potential energy of the particle is plotted as the red curve versus its position d. The minimal kinetic energy of the particle is reached at x 3 e. None of the above.
Lecture 19: More on Energy Applications
Reading a Potential Energy Curve the other way: Force from Potential Energy Potential energy (change) and force are related by U= Fdx. Take the derivative of the latter to obtain the force as the derivative of the potential energy: F = du/dx Based on this observation, what is(are) the point(s) at which the particle of high energy in the simulation experiences zero net force? Group work: Plot the Force
Another Look at the Example Consider the Block and Spring System simulation. What percentage of the total mechanical energy is stored in the spring, i.e. is potential energy, at time 1/3 T, assuming the block started at t=0 at x = x max, and T is the period of the motion, i.e. the time it takes the block to go from x= x max via x = 0 and x = x max back to x = x max (or any other round trip )? X= x % Answer: x = cos 2 (2π/T*1/3 T)*100 = 0.25*100 = 25
Example (first motion of Energy Bar Chart exercise) Ball falling from height h=h to height h=g, define U to be zero at floor (h=0), i.e. U(0)=0 What is the velocity of the ball at G? Use Energy bar chart to set up equation ½mv i2 + mgh i = ½mv f2 + mgh f Unknown: v f2 = v i2 +2g(h i - h f ) Note: is good old kinematics formula! (double sign!) Numerical example: take H = 3m, G =1m v f2 = 0 +19.6m/s 2 (2m) v f = - 6.26 m/s (sign from )
Mechanical Energy & Nonconservative forces If all forces are conservative, mechanical energy is conserved: K i +U i = K f +U f If there are non-conservative (dissipative) forces, we must calculate the work they do and add it to the left side: K i +U i +W by nc forces = K f +U f Skateboard Example: Friction does W non-conservative = F friction d Note that d is the unknown
Group Work Draw an energy bar chart for the following process: A mass on the horizontal spring starts out in the compressed position, then is let go (not attached to spring). The track is frictionless from the equilibrium point of the spring onward, but has friction before that.
Example Worksheet A 60kg skateboarder starts up a 20 degree slope at 5m/s, then falls and slides up the hill on his kneepads. The coefficient of friction is 0.30. How far does he slide before stopping? K i +U i +W by nc forces = K f +U f Friction does W non-conservative = F friction d Note that d is the unknown
Group Work Draw an energy bar chart for the following process: A mass on a horizontal frictionless table is pushed by a spring. It starts out in the compressed position, then is let go. Mass is not attached to spring