1. Applications & Examples of Newton’s Laws

2. Forces are VECTORS!! Newton’s 2 nd Law: ∑F = ma ∑F = VECTOR SUM of all forces on mass m  Need VECTOR addition to add forces in the 2 nd Law! –Forces add according to rules of VECTOR ADDITION! (Ch. 3)

3. Newton’s 2 nd Law problems: STEP 1: Sketch the situation!! –Draw a “Free Body” diagram for EACH body in problem & draw ALL forces acting on it. Part of your grade on exam & quiz problems! STEP 2: Resolve the forces on each body into components –Use a convenient choice of x,y axes Use the rules for finding vector components from Ch. 3.

4. STEP 3: Apply Newton’s 2nd Law to EACH BODY SEPARATELY : ∑F = ma –A SEPARATE equation like this for each body! –Resolved into components: ∑F x = ma x ∑F y = ma y Notice that this is the LAST step, NOT the first!

5. Conceptual Example Moving at constant v, with NO friction, which free body diagram is correct?

6. Example Particle in Equilibrium “Equilibrium” ≡ The total force is zero. ∑F = 0 or ∑F x = 0 & ∑F y = 0 Example (a) Hanging lamp (massless chain). (b) Free body diagram for lamp. ∑F y = 0  T – F g = 0; T = F g = mg (c) Free body diagram for chain. ∑F y = 0  T – T´ = 0; T ´ = T = mg

7. Example Particle Under a Net Force Example (a) Crate being pulled to right across a floor. (b) Free body diagram for crate. ∑F x = T = ma x  a x = (T/m) a y = 0, because of no vertical motion. ∑F y = 0  n – F g = 0; n = F g = mg

8. Example Normal Force Again “Normal Force” ≡ When a mass is in contact with a surface, the Normal Force n = force perpendicular to (normal to) the surface acting on the mass. Example Book on a table. Hand pushing down. Book free body diagram.  a y = 0, because of no vertical motion (equilibrium). ∑F y = 0  n – F g - F = 0  n = F g + F = mg + F Showing again that the normal force is not always = & opposite to the weight!!

9. Example A box of mass m = 10 kg is pulled by an attached cord along a horizontal smooth (frictionless!) surface of a table. The force exerted is F P = 40.0 N at a 30.0° angle as shown. Calculate: a. The acceleration of the box. b. The magnitude of the upward normal force F N exerted by the table on the box. Free Body Diagram The normal force, F N is NOT always equal & opposite to the weight!!

10. Two boxes are connected by a lightweight (massless!) cord & are resting on a smooth (frictionless!) table. The masses are m A = 10 kg & m B = 12 kg. A horizontal force F P = 40 N is applied to m A. Calculate: a. The acceleration of the boxes. b. The tension in the cord connecting the boxes. Example Free Body Diagrams

11. Example 5.4 : Traffic Light at Equilibrium (a) Traffic Light, F g = mg = 122 N hangs from a cable, fastened to a support. Upper cables are weaker than vertical one. Will break if tension exceeds 100 N. Does light fall or stay hanging? (b) Free body diagram for light. a y = 0, no vertical motion. ∑F y = 0  T 3 – F g = 0 T 3 = F g = mg = 122 N (c) Free body diagram for cable junction (zero mass). T 1x = -T 1 cos(37°), T 1y = T 1 sin(37°) T 2x = T 2 cos(53°), T 2y = T 2 sin(53°), a x = a y = 0. Unknowns are T 1 & T 2. ∑F x = 0  T 1x + T 2x = 0 or -T 1 cos(37°) + T 2 cos(53°) = 0 (1) ∑F y = 0  T 1y + T 2y – T 3 = 0 or T 1 sin(37°) + T 2 sin(53°) – 122 N = 0 (2) (1) & (2) are 2 equations, 2 unknowns. Algebra is required to solve for T 1 & T 2 ! Solution: T 1 = 73.4 N, T 2 = 97.4 N

12. Example 5.6: Runaway Car

13. Example 5.7: One Block Pushes Another

14. Example 5.8: Weighing a Fish in an Elevator

15. Example 5.9: Atwood Machine

16. Example 4-13 (“Atwood’s Machine”) Two masses suspended over a (massless frictionless ) pulley by a flexible (massless) cable is an “Atwood’s machine”. Example: elevator & counterweight. Figure: Counterweight m C = 1000 kg. Elevator m E = 1150 kg. Calculate a. The elevator’s acceleration. b. The tension in the cable. a E = - a a C = a aa aa Free Body Diagrams

17. Conceptual Example mg = 2000 N Advantage of a Pulley A mover is trying to lift a piano (slowly) up to a second- story apartment. He uses a rope looped over 2 pulleys. What force must he exert on the rope to slowly lift the piano’s mg = 2000 N weight? Free Body Diagram

18. Example: Accelerometer A small mass m hangs from a thin string & can swing like a pendulum. You attach it above the window of your car as shown. What angle does the string make a. When the car accelerates at a constant a = 1.20 m/s 2 ? b. When the car moves at constant velocity, v = 90 km/h? Free Body Diagram

19. Example = 300 N F T2x = F T cosθ F T2y = -F T sinθ F T1x = -F T cosθ F T1y = -F T sinθ Free Body Diagram

20. Inclined Plane Problems Understand ∑F = ma & how to resolve it into x,y components in the tilted coordinate system!! Engineers & scientists MUST understand these! a The tilted coordinate System is convenient, but not necessary.

21. A box of mass m is placed on a smooth (frictionless!) incline that makes an angle θ with the horizontal. Calculate: a. The normal force on the box. b. The box’s acceleration. c. Evaluate both for m = 10 kg & θ = 30º Example: Sliding Down An Incline Free Body Diagram

22. Example 5.10 Inclined Plane, 2 Connected Objects