Document Sample

Chapter 5 Applications of Newton’s Laws We shall apply Newton’s laws of motion to a variety of situations in which several types of forces are involved: • Gravitational force • Tension • Contact Forces (including friction) (5-1) Gravitational force Fg Also known as the “weight” of m Fg an object g earth All objects near the surface of the earth are uniformly accelerated towards the center of the earth. The magnitude of the acceleration g = 9.8 m/s 2 From Newton’s second law we get that Fg = ma = mg (5-2) A practical method of measuring Fg Use a spring scale as shown in the figure. y-axis The spring has a pointer attached to it so that we can determine its elongation System = grapes. There are two forces acting on the grapes: The gravitational force mg and the spring force Fs which is given by: Fs = k|y|. Here y is the elongation of the spring The net force Fynet = Fs - mg = may = 0 because the grapes are stationary. Thus mg = Fs = k|y| (5.3) T Tension crate rope It is the force exerted by a rope or a floor cable Tension has the following characteristics: • Tension always pulls (never pushes) • The direction of T is along the rope pulley • The magnitude |T| is constant T even when T changes direction T (as in the case of the pulley) crate pull (5-4) Normal Force (contact force) Characteristics of the normal force FN • FN is perpendicular to the contact surface • FN always pushes away from the contact surface (never pulls) FN crate Fg floor (5-5) The normal force and Newton’s third law FN = Fcf System = crate crate The normal force on the crate exerted by the floor is labeled Fcf floor System = floor crate The normal force on the floor exerted by the floor is labeled F fc floor Ffc Newton’s third law: Fcf + F fc = 0 → Ffc = - Fcf = - FN The crate exerts a force F fc on the floor which is equal and opposite to FN (5-6) Q: How can we measure F N ? A: With a bathroom scale! The bathroom scale measures the normal force (and not necessarily mg unless the object on the scale does not accelerate) (5-7) Fas is the normal force exerted on the apple by the scale. apple FN = Fas Fsa is the normal force exerted on the scale by the apple scale These two forces are equal and FN = Fsa opposite by virtue of Newton’s third law y-axis System = apple Fynet = FN - mg FN Fynet = may = 0 → mg FN - mg = 0 → (5-8) Indeed: FN = mg What does the scale reads when the y-axis elevator elevator accelerates upwards ? System = man .. Fynet = FN - mg Fynet = ma (Newton’s second law) FN → FN - mg = ma → FN = mg + ma a mg FN > mg The scale in this case gives a reading which is higher than the actual weight scale mg (5-9) What does the scale reads when the y-axis elevator elevator accelerates downwards ? System = man .. Fynet = FN - mg Fynet = - ma (Newton’s second law) FN → FN - mg = - ma → FN = mg - ma a mg FN < mg The scale in this case gives a reading which is lower than the actual weight scale mg (5-10) Atwood’s Machine a It consists of two masses m1 and a m2 (m2 > m 1) connected with a y rope that goes around a pulley. The rope and the pulley have m2 negligible masses. m1 The two masses are released and move under gravity which acts downwards. The heavier mass m 2 will move down and the lighter mass m1 will move up. Determine the acceleration a of the two masses (5-11) y a a T T a a y y m1 m2 m1g m2g m2 m1 System = m1 System = m2 Fynet = T − m1 g = m1a → Fynet = T − m2 g = − m2a → T = m1g + m1a eqs.1 T = m2 g − m2a eqs.2 We combine eqs.1 and eqs.2 → m1 g + m1a = m2 g − m2 a Solve for a ( m2 − m1 ) → a=g ( m2 + m1 ) (5-12) Friction Friction opposes motion or impending motion (5-13) (5-14) motion Friction was first studied by FN Leonardo da Vinci crate T He discovered that friction has the f rope following characteristics: floor mg • Friction opposes motion or impending motion • Static and kinetic friction are independent of contact area • The static and kinetic friction are proportional to FN Static friction fs 0 < f S < µ S FN µs is a constant known as “coefficient of static friction” Kinetic friction fk f k = µ k FN µk is a constant known as “coefficient of kinetic friction” Leonardo da Vinci (self portrait) (1452-1519) motion Consider the following experiment: FN T We start pulling a heavy crate using crate a rope with a force T that increases f rope with time. At some instant tm the floor mg crate will start to move f µk < µs We then plot the frictional force f as function of time. The static µ s FN friction increases with time. At tm the static friction becomes µ k FN maximum (µs FN) and the crate Static kinetic friction friction starts to move. The kinetic tm friction has a smaller constant Motion t value of µk FN . (5-15) starts Example (5-6),page 121 y A mass m is at rest on a x horizontal ramp. The ramp is slowly raised. Find the critical angle θc at which the mass will start to slide Fynet = FN - mg cos θ = ma y = 0 → FN = mg cos θ (eqs.1) Fxnet = mg sin θ - f S = max = 0 → mg sin θ = f S = µ S FN (eqs.2) From eqs.1 substitute FN into eqs.2 → mg sin θ = µ S mg cos θ → tan θ C = µ S (5-16) Example (5-8) page 123. An automobile has a mass m = 1000 kg. The static friction with the road is µs = 0.8 Determine the maximum forward acceleration (without spinning the wheels) Fynet = FN - mg = may = 0 → FN = mg (eqs.1) Fxnet = fs = ma → a = fs /m amax = (fs)max /m = µs FN /m (eqs.2) From eqs.1 substitute FN into eqs.2 → amax = µsmg /m amax = µs g = 0.8×9.8 = 7.8 m/s2 When the rope breaks the book does not stay on its circular path but instead flies along the tangent to the orbit Conclusion: For an object to be able to move on a circular orbit a force that points towards the center of the circle is required. This force is known as the “centripetal” force. When the centripetal force stops acting the object cannot stay on its circular orbit (5-18) (5-19) m Consider an object of mass m m moving on a circle of radius r with uniform speed v. The acceleration points towards the center of the orbit and has magnitude a = v2/r = ω2r m From Newton’s second law a force F = ma must act on the object . Thus the centripetal force F is given by the expression: mv 2 F= or F = mω 2r r The magnitude of F is constant, its direction is not. It always points towards the center of the circular orbit. Recipe for problems that involve uniform circular motion of an C . y R object of mass m on a circular orbit of radius R with speed v x m v • Draw the force diagram for the object • Choose one of the coordinate axes (the y-axis in this diagram) to point towards the orbit center C • Determine Fynet • Set Fynet = mv2/R or Fynet = mω2R (5-20) Example (5-11) page 128 A small dice sits 0.15 m from the center of a turntable. If µs between the turntable and the (5-21) dice is 0.55 what is the largest y possible angular speed such x that the dice will not slide off Fynet = FN - mg = may = 0 → FN = mg (eqs.1) Fxnet = fs = mω2r (eqs.2) → ω2 = fs /mr ω2max = (fs )max /mr From eqs.1 we have: (fs)max = µs FN Thus ω2max = µs mg/mr → ωmax = [µs g/r]1/2 = 6 rad/s Example (5-12) page 129 A space station in the form of a hollow circular tube rotates around its axis. If the distance from the station outer wall to the center is 50 m what should be y the speed v of the outer wall so that a scale will read what it would read on earth Fynet = FN = mv2/R (eqs.1) Also FN = mg (eqs.2) If we combine the two equations we get: mv2/R = mg → v = [Rg]1/2 = [50×9.8]1/2 = 22.1 m/s (5-22)

DOCUMENT INFO

Shared By:

Categories:

Tags:
newton's laws, newton's second law, the force, chapter 4, kinetic energy, normal force, potential energy, internet explorer, newton's third law, newton's first law, work done, newton's laws of motion, the normal, chapter 6, uniform circular motion

Stats:

views: | 60 |

posted: | 12/23/2009 |

language: | English |

pages: | 23 |

OTHER DOCS BY umsymums38

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.