Lecture 15

1. Lecture 15 • Chapter 11 • Employ conservative and non-conservative forces • Use the concept of power (i.e., energy per time) • Chapter 12 • Extend the particle model to rigid-bodies • Understand the equilibrium of an extended object. • Understand rotation about a fixed axis. • Employ “conservation of angular momentum” concept Goals: Assignment: • HW7 due March 25th • For Thursday: Read Chapter 12, Sections 7-11 do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia”

2. More Work: “2-D” Example (constant force) • An angled force, F= 10 N,pushes a box across a frictionless floor for a distance x= 5 m and y= 0 m • (Net) Work is Fxx= F cos(-45°) x = 50 x 0.71 Nm = 35 J • Notice that work reflects energy transfer Finish Start F q = -45° Fx x

3. Work in 3D….(assigning U to be external to the system) • x, y and z with constant F:

4. A q Ay Ax î • Calculation can be made in terms of components. A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz ) Calculation also in terms of magnitudes and relative angles. A B≡ | A | | B | cosq A tool: Scalar Product (or Dot Product) A·B≡ |A| |B| cos(q) • Useful for performing projections. A î= Ax î  î = 1 î j = 0 You choose the way that works best for you!

5. Scalar Product (or Dot Product) Compare: A B = (Ax )(Bx) + (Ay )(By ) + (Az )(Bz ) Redefine AF (force), B Dr (displacement) Notice: F Dr = (Fx )(Dx) + (Fy )(Dz ) + (Fz )(Dz) So here F Dr= W More generally a Force acting over a Distance does Work

6. “Scalar or Dot Product” Definition of Work, The basics Ingredients:Force (F ), displacement ( r) Work, W, of a constant force F acts through a displacement  r: W = F· r(Work is a scalar) F  r  displacement If we know the angle the force makes with the path, the dot product gives usF cos qandDr If the path is curved at each point and

7. v ExerciseWork in the presence of friction and non-contact forces • A box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below. • How many forces (including non-contact ones) are doing work on the box ? • Of these which are positive and which are negative? • State the system (here, just the box) • Use a Free Body Diagram • Compare force and path • 2 • 3 • 4 • 5

8. Exercise Work in the presence of friction and non-contact forces • A box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below. • How many forces are doing work on the box ? • And which are positive (T) and which are negative (f, mg)? (For mg only the component along the surface is relevant) • Use a Free Body Diagram (A)2 (B)3 is correct • 4 • 5 v N T f mg

9. Work and Varying Forces (1D) Area = Fx Dx F is increasing Here W = F·r becomes dW = Fdx • Consider a varying force F(x) Fx x Dx Finish Start F F q = 0° Dx Work has units of energy and is a scalar!

10. Example: Hooke’s Law Spring (xi equilibrium) ti vi m spring at an equilibrium position x V=0 F t m spring compressed • How much will the spring compress (i.e. x = xf - xi) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vi) on frictionless surface as shown below and xi is the equilibrium position of the spring?

11. ti vi m spring at an equilibrium position x F V=0 t m spring compressed Work signs Notice that the spring force is opposite the displacement For the mass m, work is negative For the spring, work is positive They are opposite, and equal (spring is conservative)

12. Uf rf rf ò U = Uf - Ui = - W = -F•dr ri Ui ri Conservative Forces & Potential Energy • For any conservative force F we can define a potential energy function U in the following way: The work done by a conservative force is equal and opposite to the change in the potential energy function. • This can be written as: ò W=F·dr= - U

13. Conservative Forces and Potential Energy • So we can also describe work and changes in potential energy (for conservative forces) DU = - W • Recalling (if 1D) W = Fx Dx • Combining these two, DU = - Fx Dx • Letting small quantities go to infinitesimals, dU = - Fx dx • Or, Fx = -dU / dx

14. h = 1 m sin 30° = 0.5 m ExerciseWork Done by Gravity • An frictionless track is at an angle of 30° with respect to the horizontal. A cart (mass 1 kg) is released from rest. It slides 1 meterdownwards along the track bounces and then slides upwards to its original position. • How much total work is done by gravity on the cart when it reaches its original position? (g = 10 m/s2) 1 meter 30° (A) 5 J (B) 10 J (C) 20 J (D) 0 J

15. Non-conservative Forces : • If the work done does not depend on the path taken, the force involved is said to be conservative. • If the work done does depend on the path taken, the force involved is said to be non-conservative. • An example of a non-conservative force is friction: • Pushing a box across the floor, the amount of work that is done by friction depends on the path taken. and work done is proportional to the length of the path !

16. Path 2 Path 1 A Non-Conservative Force, Friction • Looking down on an air-hockey table with no air flowing (m > 0). • Now compare two paths in which the puck starts out with the same speed (Ki path 1 = Ki path 2) .

17. Path 2 Path 1 A Non-Conservative Force Since path2 distance >path1 distance the puck will be traveling slower at the end of path 2. Work done by a non-conservative force irreversibly removes energy out of the “system”. Here WNC = Efinal - Einitial < 0 and reflects Ethermal

18. Work & Power: • Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the same mass. • Assuming identical friction, both engines do the same amount of work to get up the hill. • Are the cars essentially the same ? • NO. The Corvette can get up the hill quicker • It has a more powerful engine.

19. Work & Power: • Power is the rate at which work is done. • Average Power is, • Instantaneous Power is, • If force constant, W= F Dx = F (v0Dt + ½ aDt2) and P = W / Dt = F (v0 + aDt)

20. Work & Power: • Power is the rate at which work is done. Units (SI) are Watts (W): Instantaneous Power: Average Power: 1 W = 1 J / 1s Example: • A person of mass 80.0 kg walks up to 3rd floor (12.0m). If he/she climbs in 20.0 sec what is the average power used. • Pavg = F h / t = mgh / t = 80.0 x 9.80 x 12.0 / 20.0 W • P = 470. W

21. Power time Power Z3 time Power time Exercise Work & Power • Top • Middle • Bottom • Starting from rest, a car drives up a hill at constant acceleration and then suddenly stops at the top. • The instantaneous power delivered by the engine during this drive looks like which of the following,

22. Chap. 12: Rotational Dynamics • Up until now rotation has been only in terms of circular motion with ac = v2 / R and | aT | = d| v | / dt • Rotation is common in the world around us. • Many ideas developed for translational motion are transferable.

23. Conservation of angular momentum has consequences How does one describe rotation (magnitude and direction)?

24. Rotational Dynamics: A child’s toy, a physics playground or a student’s nightmare • A merry-go-round is spinning and we run and jump on it. What does it do? • We are standing on the rim and our “friends” spin it faster. What happens to us? • We are standing on the rim a walk towards the center. Does anything change?

25. Overview (with comparison to 1-D kinematics) Angular Linear And for a point at a distance R from the rotation axis: x = R v =  R aT =  R Here aT refers to tangential acceleration

26. Lecture 15 Assignment: • HW7 due March 25th • For Thursday: Read Chapter 12, Sections 7-11 do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia”