Conservation of Mechanical Energy 8.01

Transcription

1 Conservation o Mechanical Energy 8.01

2 Non-Conservative Forces Work done on the object by the orce depends on the path taken by the object Example: riction on an object moving on a level surace F riction = µ k N W riction =!F riction "x =!µ k N"x < 0

3 Non-Conservative Forces Deinition: Non-conservative orce Whenever the work done by a orce in moving an object rom an initial point to a inal point depends on the path, then the orce is called a non-conservative orce.

4 Change in Energy or Conservative and Non-conservative Forces Total orce: r r r F total = F total + F total c nc Total work done is change in kinetic energy: B B total r total total r total W total = $ F r r r! dr = $(F c + F nc )! dr = "# U +W nc = #K A A Energy Change:!K +!U total = W nc

5 Checkpoint Problem: Cart- Spring on an Inclined Plane An object o mass m slides down a plane that is inclined at an angle θ rom the horizontal. The object starts out at rest. The center o mass o the cart is a distance d rom an unstretched spring with spring constant k that lies at the bottom o the plane. a) Assume the inclined plane to be rictionless. How ar will the spring compress when the mass irst comes to rest? b) Now assume that the inclined plane has a coeicient o kinetic riction µ. How ar will the spring compress when the mass irst comes to rest? How much energy has been transormed into heat due to riction?

6 1.. Reading Quiz : Cart-Spring on an Inclined Plane An object o mass m slides down a plane that is inclined at an angle θ rom the horizontal. The object starts out at rest. The center o mass o the cart is an unknown distance d rom an unstretched spring with spring constant k that lies at the bottom o the plane. Assume the inclined plane to be rictionless. The spring compress a distance x when the mass irst comes to rest? Find an expression or the distance d. 1 mg sin! d = kx " x 1 d = kx mg sin! 1 d = kx " x sin! mg mg sin! d = kx + x 1 d = kx mg 1 d = kx + x sin! mg

7 Strategy: Using Multiple Ideas Force and Energy Need second law in radial direction

8 Summary: Change in Mechanical Energy Total orce: Total work: W total = Change in potential energy: Total work done is change in kinetic energy: Mechanical Energy Change: Conclusion: inal r F total r = F total + F c r total nc inal r F total r r! dr = total r total (F + F ) r! dr " " initial initial c nc inal!u total = " $ F r total r # dr initial W total =!" U total +W = "K nc!e mechanical "! K +!U total c W =!K +!U total nc

9 Modeling the Motion using Force and Energy Concepts Force and Newton s Second Law: Draw all relevant ree body orce diagrams Identiy non-conservative orces. Calculate non-conservative work Change in Mechanical Energy: r W nc = " F nc! dr. initial inal r Choose initial and inal states and draw energy diagrams. Choose zero point P or potential energy or each interaction in which potential energy dierence is well-deined. Identiy initial and inal mechanical energy. Apply Energy Law. W nc =!K +!U total

10 Mechanical Energy Accounting Initial state: Total initial kinetic energy Total initial potential energy Total initial mechanical energy Final state: K initial = K 1,initial + K,initial +!!! U initial = U 1,initial +U,initial +!!! mechanical E initial = K initial + U initial Total inal kinetic energy Total inal potential energy Total inal mechanical energy K inal = K 1,inal + K,inal +!!! U inal = U 1,inal +U,inal +!!! Apply Energy Law: mechanical E inal = K inal + U inal mechanical W nc = E inal! E initial mechanical

11 Worked Example: Block Sliding o Hemisphere A small point like object o mass m rests on top o a sphere o radius R. The object is released rom the top o the sphere with a negligible speed and it slowly starts to slide. Find an expression or the angle θ with respect to the vertical at which the object just loses contact with the sphere.

12 Example: Energy Changes A small point like object o mass m rests on top o a sphere o radius R. The object is released rom the top o the sphere with a negligible speed and it slowly starts to slide. Find an expression or the angle θ with respect to the vertical at which the object just loses contact with the sphere. Energy Flow diagrams Initial state Final State

13 Example: Energy Changes A small point like object o mass m rests on top o a sphere o radius R. The object is released rom the top o the sphere with a negligible speed and it slowly starts to slide. Find an expression or the angle θ with respect to the vertical at which the object just loses contact with the sphere. K initial 0 K inal = 1 mv U initial = 0 U inal =!mgr(1! cos" ) mechanical mechanical 1 E initial 0 E inal = mv " mgr (1 " cos! ) mechanical mechanical W nc = 0 = E inal! E initial " 0 = 0! $ % # 1 mv! mgr(1! cos" )( & ) ' 1 mv = mgr(1! cos" )

14 Recall Modeling the Motion: Newton s Second Law Deine system, choose coordinate system. Draw orce diagram. Newton s Second Law or each direction. Example: x-direction Example: Circular motion î : F total = m d x x dt. rˆ : F total =! m v. r R

15 Example (con t): Free Body Force Diagram Newton s Second Law r ˆ : v N " mg cos! = " m R ˆ d! è : mg sin! = mr Constraint condition: dt Radial N Equation = 0! becomes =! v mg cos! = m " R Energy Condition: Conclusion: 1 mv = R mg cos! 1 mv = mgr(1! cos" ) R mgr (1 " cos! ) = mg cos! # " 1 # $ cos! = %! = cos & ' 3 ( 3 )

16 Checkpoint Problem: Loop-the- Loop An object o mass m is released rom rest at a height h above the surace o a table. The object slides along the inside o the loop-the-loop track consisting o a ramp and a circular loop o radius R shown in the igure. Assume that the track is rictionless. When the object is at the top o the track (point a) it pushes against the track with a orce equal to three times it s weight. What height was the object dropped rom?

17 Demo slide: Loop-the-Loop B95 e=demo.php?letnum=b 95&show=0 A ball rolls down an inclined track and around a vertical circle. This demonstration oers opportunity or the discussion o dynamic equilibrium and the minimum speed or sae passage o the top point o the circle.

18 Checkpoint Problem: Extreme Skier An extreme skier is accelerated rom rest by a spring-action cannon, skis once around the inside o a vertically oriented circular loop, then comes to a stop on a carpeted up-acing slope. Assume the cannon has a spring constant k and a cocked displacement x 0, the loop has a radius R, and the slope makes an angle θ to the horizontal. The only surace with riction is the carpet, represented by a riction constant µ. Gravity acts downward, with acceleration g, as shown. What is the linear distance d the skier travels on the carpet beore coming to rest?

19 Chcckpoint Problem: Block-Spring System with Friction A block o mass m slides along a horizontal surace with speed v 0. At t =0 it hits a spring with spring constant k and begins to experience a riction orce. The coeicient o riction is variable and is given by µ k = bx where b is a constant. Find how ar the spring has compressed when the block has irst come momentarily to rest.

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