Lecture 11

1. Lecture 11 Chapter 8: Employ rotational motion models with friction or in free fall Chapter 9: Momentum & Impulse • Understand what momentum is and how it relates to forces • Employ momentum conservation principles • In problems with 1D and 2D Collisions • In problems having an impulse (Force vs. time) Goals: Assignment: • Read through Chapter 10, 1st four section • MP HW6, due Wednesday 3/3

2. Zero Gravity Ride • One last reprisal of the Free Body Diagram • Remember 1 Normal Force is  to the surface 2 Friction is parallel to the contact surface 3 Radial (aka, centripetal) acceleration requires a net force

3. Zero Gravity Ride A rider in a horizontal “0 gravity ride” finds herself stuck with her back to the wall. Which diagram correctly shows the forces acting on her?

4. Banked Curves In the previous car scenario, we drew the following free body diagram for a race car going around a curve at constant speed on a flat track. Because the acceleration is radial (i.e., velocity changes in direction only) we need to modify our view of friction. n Ff mg So, what differs on a banked curve?

5. y x N mar Ff q mg Banked Curves (high speed) 1 Draw a Free Body Diagram for a banked curve. 2 Use a rotated x-y coordinates 3 Resolve into components parallel and perpendicular to bank

6. y x N q mar mar sin q mar cos q Ff q mg cos q mg sin q Banked Curves, high speed 4 Apply Newton’s 1st and 2nd Laws S Fx = -mar cos q = - Ff - mg sin q S Fy = mar sin q = 0 - mg cos q + N Friction model  Ff≤m N (maximum speed when equal)

7. y x mar Banked Curves, low speed 4 Apply Newton’s 1st and 2nd Laws N Ff q mar sin q mar cos q q mg cos q S Fx = -mar cos q = + Ff - mg sin q S Fy = mar sin q = 0 - mg cos q + N Friction model  Ff≤m N (minimum speed when equal but not less than zero!) mg sin q

8. Banked Curves, constant speed vmax = (gr)½ [ (m + tan q) / (1 - m tan q)] ½ vmin = (gr)½ [(tan q - m) / (1 + m tan q)] ½ Dry pavement “Typical” values of r = 30 m, g = 9.8 m/s2, m = 0.8, q = 20° vmax = 20 m/s (45 mph) vmin = 0 m/s (as long as m > 0.36 ) Wet Ice “Typical values” of r = 30 m, g = 9.8 m/s2, m = 0.1, q =20° vmax = 12 m/s (25 mph) vmin = 9 m/s (Ideal speed is when frictional force goes to zero)

9. Navigating a hill Knight concept exercise: A car is rolling over the top of a hill at speed v. At this instant, • n > w. • n = w. • n < w. • We can’t tell about n without knowing v. At what speed does the car lose contact? This occurs when the normal force goes to zero or, equivalently, when all the weight is used to achieve circular motion. Fc = mg = m v2 /r  v = (gr)½ (just like an object in orbit) Note this approach can also be used to estimate the maximum walking speed.

10. Orbiting satellites vT = (gr)½

11. Locomotion: how fast can a biped walk?

12. How fast can a biped walk? What about weight? A heavier person of equal height and proportions can walk faster than a lighter person A lighter person of equal height and proportions can walk faster than a heavier person To first order, size doesn’t matter

13. How fast can a biped walk? What about height? A taller person of equal weight and proportions can walk faster than a shorter person A shorter person of equal weight and proportions can walk faster than a taller person To first order, height doesn’t matter

14. How fast can a biped walk? What can we say about the walker’s acceleration if there is UCM (a smooth walker) ? Acceleration is radial ! So where does it, ar, come from? (i.e., what external forces act on the walker?) 1. Weight of walker, downwards 2. Friction with the ground, sideways

15. Impulse & Linear Momentum • Transition from forces to conservation laws Newton’s Laws  Conservation Laws Conservation Laws  Newton’s Laws They are different faces of the same physics NOTE: We have studied “impulse” and “momentum” but we have not explicitly named them as such Conservation of momentum is far more general than conservation of mechanical energy

16. Forces vs time (and space, Ch. 10) • Underlying any “new” concept in Chapter 9 is • A net force changes velocity (either magnitude or direction) • For any action there is an equal and opposite reaction • If we emphasize Newton’s 3rd Law and emphasize changes with time then this leads to the Conservation of Momentum Principle

17. 10 F (N) 0 2 Time (sec) Example • There many situations in which the sum of the product “mass times velocity” is constant over time • To each product we assign the name, “momentum” and associate it with a conservation law. (Units: kg m/s or N s) • A force applied for a certain period of time can be graphed and the area under the curve is the “impulse” Area under curve : “impulse” With: m Dv = FavgDt

18. Force curves are usually a bit different in the real world

19. 10 A on B F (N) 0 B on A -10 0 2 Time (sec) Example with Action-Reaction • Now the 10 N force from before is applied by person A on person B while standing on a frictionless surface • For the force of A on B there is an equal and opposite force of B on A MA x DVA = Area of top curve MB x DVB = Area of bottom curve Area (top) + Area (bottom) = 0

20. Example with Action-Reaction MADVA + MBDVB = 0 MA [VA(final) - VA(initial)] + MB [VB(final) - VB(initial)] = 0 Rearranging terms MAVA(final) +MB VB(final) = MAVA(initial) +MB VB(initial) which is constant regardless of M or DV (Remember: frictionless surface)

21. Example with Action-Reaction MAVA(final) +MB VB(final) = MAVA(initial) +MB VB(initial) which is constant regardless of M or DV Define MV to be the “momentum” and this is conserved in a system if and only if the system is not acted on by a netexternal force (choosing the system is key) Conservation of momentum is a special case of applying Newton’s Laws

22. Alpha Decay 238U 234Th 4He v2 v1 Applications of Momentum Conservation Radioactive decay: Explosions Collisions

23. Impulse & Linear Momentum • Definition: For a single particle, the momentum p is defined as: (p is a vector since v is a vector) p≡ mv So px = mvxand so on (y and z directions) • Newton’s 2nd Law: F = ma • This is the most general statement of Newton’s 2nd Law

24. Momentum Conservation • Momentum conservation (recasts Newton’s 2nd Law when net external F = 0) is an important principle • It is a vector expression (Px, Py and Pz) . • And applies to any situation in which there is NO net external force applied (in terms of the x, y & z axes).

25. Momentum Conservation • Many problems can be addressed through momentum conservation even if other physical quantities (e.g. mechanical energy) are not conserved • Momentum is a vector quantity and we can independently assess its conservation in the x, y and z directions (e.g., net forces in the z direction do not affect the momentum of the x & y directions)

26. 2 1 Exercise 2Momentum Conservation • Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface. • The ball hitting box 1 bounces elastically back, while the ball hitting box 2 sticks. • Which box ends up moving fastest ? • Box 1 • Box 2 • same

27. Lecture 11 Assignment: • For Monday: Read through Chapter 10, 1st four sections • MP HW6 due Wednesday 3/3