Conservation of Mechanical Energy
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College Preparatory Program • Saudi Aramco Conservation of Mechanical Energy Conservation of Mechanical Energy To see if you grasped the concepts try this problem first on your own: A block is pushed back 20 cm to compress a spring with a spring constant k = 500 N/m. The block is then released; it slides down a frictionless ramp and collides with a block moving 7 m/s to the left on a flat surface. 3ππ ππ π 7π/π 2ππ a) b) c) d) Find the velocity of the block right before it slides down the incline Find the velocity halfway down the ramp Find the velocity at the bottom of the incline (π¦ = 0) If the two blocks stick together, find the velocity of the blocks just after they collide SOLUTION a) Are the forces conservative or not? Gravitational and Spring Forces are conservative If the forces are conservative, the total mechanical energy is conserved π¬π = π¬π You can also say βπ¬ = π Friction and other forces are not conservative If the forces are not conservative, the total mechanical energy is NOT conserved πΎπππππ ππππππ = βπ¬ = π¬π − π¬π College Preparatory Program • Saudi Aramco Conservation of Mechanical Energy Since there is no friction, you will use the concept of conservation of energy. All you need to do now is find the expressions of πΈ1 and of πΈ2 and balance them. E is the sum of potential π and the kinetic energy πΎ πΈ = π + πΎ At the time of the release, what is the total mechanical energyπΈ1 ? π1 = ππ + ππ πΎ1 = 0 Released from rest so π£ 1 = 0 and πΎ1 = 0 Elastic potential energy ππ because the spring is compressed Gravitational potential energy ππ 1 ππ = πππ ππ = 2kπ₯ 2 Therefore: 1 πΈ1 = 2kπ₯ 2 + πππ Right before the block slides down the incline, what’s the total mechanical energy πΈ2 ? 1 π2 = ππ = πππ πΎ2 = 2 ππ£22 1 So πΈ2 = πππ + 2 ππ£22 College Preparatory Program • Saudi Aramco Conservation of Mechanical Energy Now balance the energies: π¬π = π¬π 1 2 1 2 1 kπ₯ 2 + πππ = πππ + 2 ππ£22 1 500(0.2)2 = 2 3π£22 It follows then that π£2 = 2.58 π/π Halfway down the ramp, what’s the total mechanical energy πΈ3 ? b) 1 π3 = ππ = ππ(π/2) πΎ3 = 2 ππ£32 π 1 πΈ3 = ππ( ) + ππ£32 2 2 Now balance the energies: you can use either π¬π = π¬π or π¬π = π¬π since all forces are conservative and the mechanical energy is conserved. π 1 1 ππ(2 ) + 2 ππ£32 = πππ + 2 ππ£22 1 2 π 1 ππ£32 = ππ(2 ) + 2 ππ£22 π£32 = ππ + π£22 π£32 = 9.8 × 30 + 2.582 Yielding to π£3 = 17.34 π/π which is greater than π£2 since more potential energy was transformed into kinetic energy! c) At the bottom of the incline, what’s the total mechanical energy πΈ4 ? π4 = 0 1 πΎ4 = 2 ππ£42 No gravitational potential energy since π¦ = 0. Mechanical Energy is purely Kinetic College Preparatory Program • Saudi Aramco Conservation of Mechanical Energy πΈ4 = 1 ππ£42 2 Now balance the energies: you can use either π¬π = π¬π or π¬π = π¬π or even π¬π = π¬π since all forces are conservative and the mechanical energy is conserved: 1 2 π 1 ππ£42 = ππ(2 ) + 2 ππ£32 π£42 = ππ + π£32 π£4 = 24.38 π/π which is greater than π£3 since all potential energy was transformed into kinetic energy! d) This is a collision question so you’ll need to use the concept of conservation of linear momentum. What is the total linear momentum of the system of the two blocks just before the collision? ′ πππππππ = π1 π£ππππππ + π2 π£ππππππ Mass of Block 1; π1 = 3ππ Mass of Block 2; π2 = 2ππ Velocity of Block 1 just before the collision is 24.38 π/π Velocity of Block 2 just before the collision is −7 π/π . The minus sign comes from the fact that Block 2 is moving in the opposite direction! College Preparatory Program • Saudi Aramco Conservation of Mechanical Energy πππππππ = 3 × 24.38 − 2 × 7 = 59.14 ππ. π/π What is the total linear momentum of the system of the two blocks just after the collision? ππππ‘ππ = (π1 + π2 )π£πππ‘ππ The 2 Blocks stick together Common velocity of the 2 Blocks which became one object: inelastic collision ππππ‘ππ = 5π£πππ‘ππ ππππ‘ππ = πππππππ 5π£πππ‘ππ = 59.14 π£πππ‘ππ = 11.82 π/π Now, here is an extra question to think about: is the kinetic energy of the system conserved or not? Why or why not? ο For a quick and efficient review of the concepts, click on the following: Mechanical Energy ο For more questions on conservation of energy visit, solve all the questions at: http://www.physics247.com/physics-homework-help/conservation-energy.php ο For the best resources go to: http://cpcapphysics.wikispaces.com/
