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Conservation of Mechanical Energy We have seen: W1 → 2 = U 1 − U 2 But, K1 + W1 → 2 = K 2 Implies, Rearrange, Conservation of Energy: The SUM of ALL energies remains constant. Conservation of Mechanical Energy: The SUM of ALL Potential and Kinetic energies remains constant. • Works only with conservative forces. K1 + U 1 − U 2 = K 2 Law of Conservation of Energy • Ignores thermodynamic energy, strain energy, etc. The total energy is neither increased nor decreased in any process. Energy can be transformed from one form to another, and from one body to another, but the total amount remains constant. ∴ ΔK + ΔU + Δ[all other forms of energy ]= 0 • Includes thermodynamic energy, strain energy, chemical energy, etc. K1 + U 1 = K 2 + U 2 ⇒ K1 − K 2 + U 1 − U 2 = 0 ∴ ΔK + Δ U = 0 Given: A 75.0 kg parachutists jumps off a training tower that is 85.0 m high. He lands with a vertical speed of 5.00 m/s. Assume the drag force is constant Potential Energy • The amount of work that a conservative force would do if the force were to move from its current position to a given (defined) reference position. • The reference position is called the datum. • The symbol of Potential Energy is U. • Types of forces that have Potential Energy: Required: Calculate the drag force and compare it to the man’s weight. Solution: • • Weight (i.e., the force due to gravity). • Spring. • ANY Constant Conservative Force. Friction and ALL other Non-Conservative forces DO NOT have a Potential Energy. Where did the “lost work” go? 1 Newton’s Law of Universal Gravitation Scalar form mm F = G 12 2 r21 Vector form mm F12 = − G 1 2 2 r̂21 r21 Where G = universal gravitational constant = 6.67 x 10-11 N⋅m2/kg m1 = mass of body 1 m2 = mass of body 2 r21 = position vector from body 2 to body 1 r21 = magnitude of the position vector from body 2 to body 1 r̂21 = position vector from body 2 to body 1 Escape Speed At what radial speed will an object “escape” the pull of Earth’s gravitation? Gravitational Potential Locate the datum at ∞ ∞ ∞ U (r ) = ∫ − Fdr = − ∫ G r r mmE dr r2 ∞ 1 dr 2 r r U (r ) = −GmmE ∫ ∞ 1⎤ ⎡1 ⎤ ⎡1 U (r ) = −GmmE ⎢ ⎥ = −GmmE ⎢ − ⎥ r r ∞ ⎣ ⎦r ⎣ ⎦ U (r ) = −G mmE r Impact Speed Astronomers discover a meteorite at a distance of 80,000 km from the center of Earth. The meteorite is moving directly toward Earth with a velocity of 2000 m/s What will be the velocity of the meteorite when it hits Earth’s surface? Ignore all drag effects. 2