Applications of Newton’s Laws of Motion

1. Applications of Newton’s Laws of Motion Equilibrium: no net force • Draw a simple diagram • Draw a free body diagram for one body • Identify interactions (force magnitudes and directions on that body) • Choose coordinates and determine the components of each force • Set algebraic sum of all x-components to zero to obtain an algebraic relation. In a separate equation set all the y-components equal to zero. • Repeat 2-5 for any other bodies in the problem. • Use all the simultaneous equations from the step(s) 5 to solve for the unknowns.

2. A 500 N gymnast begins climbing a 100 N rope. • what is the tension at the bottom of the rope? • what is the tension in the top of the rope?

3. A car engine of weight w is suspended from a chain linked to two other chains as shown. One chain is fastened to the ceiling and makes a 60° angle with respect to the ceiling. The other is horizontal and attached to the wall. What is the tension in the chains in terms of the weight of the engine? 60o

4. A car of weight w rests on a ramp and is held in place by a cable, once the car’s brakes and transmission are released. If the ramp is at an angle of a with respect to the horizontal, determine the tension in the cable in terms of the car’s weight. a

5. Tension over a frictionless pulley: A weight w1 slides on a frictionless 15o slope. A cable attached to the weight is also attached over a pulley to a counterweight w2 chosen to just counter balance w1, so that the entire system moves at constant velocity. Determine w2 in terms of w1. w1 15o w2

6. Dynamics: • Draw a simple diagram • Draw a free body diagram for one body • Identify interactions (force magnitudes and directions on that body) • Choose coordinates and determine the components of each force • Set algebraic sum of all x-components to max to obtain an algebraic relation. In a separate equation set all the y-components equal to may. • Repeat 2-5 for any other bodies in the problem. • Use all the simultaneous equations from the step(s) 5 to solve for the unknowns. deja vue

7. An iceboat (mass including rider is 200 kg) is initially at rest on a horizontal frictionless surface. What force must be applied to the boat to accelerate it to a speed of 6.00 m/s in 4.00 s? An iceboat (mass including rider is 200 kg) is initially at rest on a horizontal surface. What force must be applied to the boat to accelerate it to a speed of 6.00 m/s in 4.00 s if the opposing frictional force has a constant magnitude of 100 N?

8. An elevator and its load have a total mass of 800 kg. The elevator is originally moving downward at 10.0 m/s. The elevator is brought to rest over a distance of 25.0 m. What is the tension in the supporting cable? A 50.0 kg woman stands on a scale while riding in the elevator in the previous example. What is the reading on the scale while the elevator is coming to a stop? What is the reading once the elevator is at rest?

9. A sled rides down a frictionless surface which makes an anglea with respect to the horizontal. What is the acceleration of the sled? A 4.0 kg cart riding on a horizontal frictionless track is attached to a .500 kg rope. If a 9.00 N force is applied to the other end of the rope, determine the acceleration of the rope and cart and the tension in both ends of the rope.

10. m1 m2 A mass m1 slides on a frictionless horizontal surface. A mass less cable attached to m1 is also attached over a pulley to a counterweight m2. Determine the acceleration of the masses and the tension in the cable.

11. A simple accelerometer is made by suspending a weight by a string from the center of a level protractor. Given the mass m of the suspended object and the angle b that the string makes with respect to the vertical, determine the acceleration a. b

12. Friction • opposes motion • due to surfaces sticking together • Kinetic Friction: surfaces are moving relative to each other • a.k.a. Sliding Friction • Static Friction: surfaces are not moving relative to each other. • Static Friction prevents stationary objects from moving until sufficient force has been applied. • Rolling Friction, etc Force of friction Applied Force

13. Coefficient of Friction • Frictional forces depend upon • how hard the surfaces are being pressed together • -> force perpendicular to the surface = normal force (FN) • the types of surfaces that are in contact • -> coefficient of friction Material static: ms kinetic: mk wood on wood 0.5 0.3 wood on stone 0.5 0.4 steel on steel (smooth) 0.74 0.57 rubber tire on dry concrete 1.0 0.8 rubber tire on wet concrete 0.3 0.25 steel on Teflon 0.04 0.04 see also table 5-1

14. A 500 N crate is dragged by a rope across a horizontal surface. To start it moving takes 230 N of horizontal force. Once moving, it takes 200 N to keep it moving at constant speed. What are the coefficients of friction? What is the friction force if the crate is at rest and a horizontal force of 50 N is applied to it? What is the force necessary keep the crate moving at constant velocity if the rope is pulled at an angle of 30o above the horizontal?

15. A chute is being built along which crates are to be slid down at constant speed. The coefficient of kinetic friction is mk. • What angle should the chute make with respect to the horizontal? • What is the acceleration of a moving crate if the angle is actually greater than this critical angle? q

16. Fluid resistance: force on a body moving through a fluid (gas/liquid) • always opposes motion • low speeds: force is proportional to speed fv • f= kv : k depends upon fluid, geometric size/shape of body • high speeds: force is proportional to speed2fv2 • f= Dv2 : D depends upon fluid, geometric size/shape of body • terminal velocity:

17. Dynamics of Circular Motion • recall • now apply Frad = m arad A .300 kg mass revolves uniformly on a frictionless horizontal surface. The mass is attached to a .140 m cord tied to a pin at the center of the circle. What is the tension in the string if the mass makes 2.00 revolutions per second?

18. A conical pendulum consists of a mass m suspended from a fixed length L of string. The mass swings in a horizontal circle so that the string always makes an angle b with respect to the vertical. How are the speed and the period related to the parameters specified above? b

19. phys A circular curve of radius R is to have a flat roadbed. The coefficient of static friction between the tires and the road is ms. What is the maximum speed vmax, the car can go around the corner without slipping?

20. phys A circular curve of radius R is to have a banked roadbed so that for a speed v, the car will not need any force of friction to make the curve. Relate the angle of bank to the speed of the car and radius of the curve.

21. minimum speed r h = 2r speed decreasing speed increasing At the bottom: maximum speed • Motion in a Vertical Circle • mass on a string At the top: Critical speed: speed at top at which string goes slack (T=0) + applications to amusement park physics

22. Fundamental Forces • Gravitation (interaction between masses) • Electromagnetism (interaction between charges) • unified force: electrostatics + magnetism + … • for interacting protons, 35 orders of magnitude stronger than gravity • Strong Interaction (aka the Strong Nuclear force) • short range • Weak Interaction (aka the Weak Nuclear force) • further successful unification: Electroweak theory • Strong + Electroweak = Grand Unified Theory (GUT) • all four unified = Theory of Everything (TOE)