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LECTURE 18 MECHANICAL ENERGY Instructor: Kazumi Tolich Lecture 18 2 ¨ Reading chapter 8-3 ¤ Mechanical energy ¤ Conservation of mechanical energy Mechanical energy 3 ¨ The mechanical energy of a system is the sum of kinetic energy and potential energy of the system. Emech ≡ K sys + U sys Conservation of mechanical energy 4 ¨ If there is no external work done on the system, and there is no work done by non-conservative force, mechanical energy is conserved. From work energy theorem, Wtotal = ΔK If only internal conservaive forces are involved, Wtotal = Wc From the definition of potential energy, Wc = −ΔU ΔK = −ΔU ΔK + ΔU = 0 K i + Ui = K f + U f Conservation of mechanical energy, graphically 5 Emech Emech Demo 1 6 ¨ x2 Energy Dependence of a Spring ¤ Demonstration of elastic potential energy converted into kinetic energy and gravitational potential energy. ¤ Conservation of mechanical energy U spring 1 2 = kx 2 1 2 → U gravity + K = mgh + mv 2 Demo 2 7 Energy Well Track ¨ Triple Track ¨ ¤ Demonstration of conservation of mechanical energy Pendulum & conservation of mechanical energy 8 ¨ Swinging pendulum (neglecting air resistance and friction at the pivot) ¤ The bob-Earth system has the maximum gravitational potential energy, and has zero kinetic energy at the highest points. ¤ It has the minimum gravitational potential energy, and the maximum kinetic energy at the lowest point. Problems solving 9 ¨ Remember that you can set gravitational potential energy to be zero anywhere you want! Clicker question: 1 10 Example: 1 11 ¨ A block of mass m = 1.6 kg slides with an initial speed of v0 = 0.950 m/s on a frictionless, horizontal surface until it encounters a spring with a force constant of k = 902 N/m. The block comes to rest after compressing the spring by x = 4.00 cm. Find the spring potential energy U, kinetic energy of the block K, and the total mechanical energy of the block-spring system E for compressions of 0.00 cm, 1.00 cm, and 4.00 cm. Example: 2 12 ¨ At an amusement park, a swimmer uses a water slide to enter the main pool. If the swimmer starts at rest, slides without friction, and descends through a vertical height of h = 2.31 m, what is her speed at the bottom of the slide? Example: 3 13 ¨ Suppose the pendulum bob in the figure has a mass of m = 0.33 kg and is moving to the right at point B with a speed of vB = 2.4 m/s. Air resistance is negligible. a) What is the change in gravitational potential energy of the bob-Earth system when the bob moves from point B to point A? b) What is the speed of the bob at point A? Demo: 3 14 ¨ Galileo's Pendulum ¤ Demonstration of mechanical energy Clicker question: 2 15 Example: 4 16 ¨ The two masses (m1 and m2, m1 < m2) in the Atwood’s machine are initially at the same height and moving with a speed of v0 with m2 moving upward. How high does m2 rise above its initial position before momentarily coming to rest? Evaluate the answer for m1 = 3.7 kg, m2 = 4.1 kg, and v0 = 0.20 m/s. Example: 5 17 ¨ The length of the string, L, is 120 cm. The distance to the fixed peg, R, is 45 cm. When the ball is released from rest in the position shown, it will follow the arc shown in the figure. How fast will it be going a) when it reaches the lowest point in its swinging and b) when it reaches the highest point after the string catches on the peg?