Transcript Lecture 19
Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lecture 19 Falling with air resistance dv a g kv dt Falling with air resistance dv 2 a= = g - kv dt Terminal Velocity with Coffee Filters mg - Fr = ma where Fr is the resistance force. Fr a=gm 1. A penny and a quarter dropped from a ladder land at the same time (air resistance is negligible). 2. A coin dropped in a coffee filter from a ladder lands later than a coin without coffee filter (the terminal velocity is smaller for larger cross-section area). 3. A quarter dropped in a coffee filter will land faster than a penny in a coffee filter (the terminal velocity is larger for larger mass) 4. Two identical coins dropped in coffee filters of different diameters land at different times (the terminal velocity is smaller for larger cross-section area). Resistance force: Fr = gAv 2 A – area of the projectile For a spherical projectile in air at g = 0.25 N ´ s /m STP: 2 4 Terminal velocity: Fr a=g=0 m Fr = mg gAv = mg mg vT = gA 2 A 70-kg man with a parachute: vT ~ 5 m/s A 70-kg man without a parachute: vT ~ 70 m/s James Prescott Joule b. Dec. 24, 1818, Salford, Lancashire, England d. Oct. 11, 1889, Sale, Cheshire Discovered some basic laws of electricity and thermodynamics (Joule ’ s Law and JouleThomson Law); established the basis of the Law of Conservation of Energy and The First Law of Thermodynamics Main occupation and source of funding: Brewery Was home-schooled by some of the finest scientists of his time (including John Dalton) Since childhood, he was a fearless and meticulous experimenter As a boy, he hiked in the mountains with a pistol, studying echo and the speed of sound. He sounded the depth of Lake Windermere to be 198 ft. He tortured servants with crazy experiments. Spent his honeymoon climbing in French Alps and measuring the temperature at the top and the bottom of waterfalls 1843 (Age 24): Paddle-wheel experiment: mechanical work can be converted into heat! Therefore, heat is one of the forms of energy. It led to the First Law of Thermodynamics It also showed that energy is conserved. Energy Conservation Law! Met with hostility and disbelief. It took decades before Joule’s discovery was accepted. Only one person believed Joule. It was William Thomson (later Lord Kelvin). They started working together. As a teenager, Joule was trying to replace steam engines with electric engines in his brewery. He failed, but became interested in the connections between mechanical work, heat, and electricity. In 1840, at the age of 21, he discovered “Joule’s Law”: Heat generated in a wire = Resistance x Current^2 It showed that electricity can be converted into heat! All forces are CONSERVATIVE or NON-CONSERVATIVE A force is conservative if: The work done by the force in going from r1 to r2 is independent of the path the particle follows or The work done by particle goes from closed path, back to the force when the r1 to r2 around a r1 , is zero. Non-conservative: doesn ’ t satisfy above conditions the If a force can be written as the derivative of some function, that force is conservative. 1D case: dU Fx dx U(x) is called the potential energy function for the force F If such a function exists, then the force is conservative W conservative does NOT depend on path! 1D case: dU F dx x2 W conservative Fdx x1 x2 dU dx [U ( x2 ) U ( x1 )] dx x1 W con W [U ( x2 ) U ( x1 )] con does NOT depend on path! If Fx(x) is known, you can find the potential energy function as U ( x) Fx ( x) dx C Force of gravity: F 0i mg j Potential energy function: U mgy Const Spring: F kxi 0 j 2 kx U Const Potential energy function: 2 A particle moves in one dimension under the influence of a single conservative force given by F ( x) x 3 where is a given constant. Take the potential energy reference to be at xi=0 such that: U ( xi 0) 0 and calculate the potential energy function U(x). or nc 1 2 If W 0, KE2 U 2 KE1 U1 Spring problem revisited A block of mass M is on a horizontal surface and is attached to a spring, spring constant k. If the spring is compressed an amount A and the block released from rest, how far from unstretched position will it go before stopping if there is no friction between the block and the surface? How will this answer change is the block is not attached to the spring?? Have a great day! Reading: Chapter 9 Hw: Chapter 8 problems and exercises