Simple Harmonic Motion & Elasticity Chapter 10 Elastic Potential Energy в–є What is it? п‚§ Energy that is stored in elastic materials as a result of their stretching. в–є Where п‚§ п‚§ п‚§ п‚§ п‚§ п‚§ п‚§ is it found? Rubber bands Bungee cords Trampolines Springs Bow and Arrow Guitar string Tennis Racquet HookeвЂ™s Law A spring can be stretched or compressed with a force. в–є The force by which a spring is compressed or stretched is proportional to the magnitude of the displacement (F п‚џ x). в–є HookeвЂ™s Law: в–є Felastic = -kx Where: (N/m) k = spring constant = stiffness of spring x = displacement HookeвЂ™s Law в–є What is the graphical relationship between the elastic spring force and displacement? Felastic = -kx Slope = k Displacement HookeвЂ™s Law в–є A force acting on a spring, whether stretching or compressing, is always positive. п‚§ Since the spring would prefer to be in a вЂњrelaxedвЂќ position, a negative вЂњrestoringвЂќ force will exist whenever it is deformed. п‚§ The restoring force will always attempt to bring the spring and any object attached to it back to the equilibrium position. п‚§ Hence, the restoring force is always negative. Example 1: в–є A 0.55 kg mass is attached to a vertical spring. If the spring is stretched 2.0 cm from its original position, what is the spring constant? в–є Known: m = 0.55 kg x = -2.0 cm g = 9.81 m/s2 в–є Felastic Equations: Fnet = 0 = Felastic + Fg (1) Felastic = -kx (2) Fg = -mg (3) Substituting 2 and 3 into 1 yields: k = -mg/x k = -(0.55 kg)(9.81 m/s2)/-(0.020 m) k = 270 N/m Fg Elastic Potential Energy in a Spring в–є The force exerted to put a spring in tension or compression can be used to do work. Hence the spring will have Elastic Potential Energy. в–є Analogous to kinetic energy: PEelastic = ВЅ kx2 Example 2: в–єWhat is the difference in the potential в–є A 0.55 kg mass is attached to aelastic vertical spring with energy the system when the If deflection is is a springofconstant of 270 N/m. the spring maximum4.0 in either theits positive orposition, negativewhat is stretched cm from original direction? the Elastic Potential Energy? в–є Known: m = 0.55 kg x = -4.0 cm k = 270 N/m g = 9.81 m/s2 в–є Felastic Equations: PEelastic = ВЅ kx2 PEelastic = ВЅ (270 N/m)(0.04 m)2 PEelastic = 0.22 J Fg Elastic Potential Energy в–є What is area under the curve? Displacement A = ВЅ bпѓ—h A = ВЅ xпѓ—F A = ВЅ xпѓ—kпѓ—x A = ВЅ kпѓ—x2 Which you should see equals the elastic potential energy What is Simple Harmonic Motion? в–єSimple harmonic motion exists whenever there is a restoring force acting on an object. п‚§ The restoring force acts to bring the object back to an equilibrium position where the potential energy of the system is at a minimum. Simple Harmonic Motion & Springs в–є Simple Harmonic Motion: п‚§ An oscillation around an equilibrium position will occur when an object is displaced from its equilibrium position and released. п‚§ For a spring, the restoring force F = -kx. в–є The spring is at equilibrium when it is at its relaxed length. (no restoring force) в–є Otherwise, when in tension or compression, a restoring force will exist. Simple Harmonic Motion & Springs в–є At maximum displacement (+ x): п‚§ The Elastic Potential Energy will be at a maximum п‚§ The force will be at a maximum. п‚§ The acceleration will be at a maximum. в–є At equilibrium (x = 0): п‚§ The Elastic Potential Energy will be zero п‚§ Velocity will be at a maximum. п‚§ Kinetic Energy will be at a maximum п‚§ The acceleration will be zero, as will the unbalanced restoring force. 10.3 Energy and Simple Harmonic Motion Example 3 Changing the Mass of a Simple Harmonic Oscilator A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring? 10.3 Energy and Simple Harmonic Motion E f пЂЅ Eo 1 2 mv 2 f пЂ« 1 2 1 2 ho пЂЅ пЂЅ I пЃ· f пЂ« mgh 2 f пЂ« 1 2 kh f пЂЅ 2 kh o пЂЅ mgh o 2 2 mg k пЂЁ 2 пЂЁ0 . 20 kg пЂ© 9 . 8 m s 28 N m 2 пЂ© пЂЅ 0 .14 m 1 2 mv o пЂ« 2 1 2 I пЃ· o пЂ« mgh o пЂ« 2 1 2 2 kh o Simple Harmonic Motion of Springs в–є Oscillating systems such as that of a spring follow a sinusoidal wave pattern. Harmonic Motion of Springs вЂ“ 1 в–є Harmonic Motion of Springs (Concept Simulator) в–є Frequency of Oscillation в–є For a spring oscillating system, the frequency and period of oscillation can be represented by the following equations: f пЂЅ 1 k 2пЃ° m and T пЂЅ 2пЃ° m k в–є Therefore, if the mass of the spring and the spring constant are known, we can find the frequency and period at which the spring will oscillate. п‚§ Large k and small mass equals high frequency of oscillation (A small stiff spring). Harmonic Motion & Simple The Pendulum в–є в–є Simple Pendulum: Consists of a massive object called a bob suspended by a string. Like a spring, pendulums go through simple harmonic motion as follows. T = 2ПЂв€љl/g Where: T = period l = length of pendulum string g = acceleration of gravity в–є Note: 1. 2. This formula is true for only small angles of Оё. The period of a pendulum is independent of its mass. Conservation of ME & The Pendulum In a pendulum, Potential Energy is converted into Kinetic Energy and vise-versa in a continuous repeating pattern. в–є PE = mgh KE = ВЅ mv2 MET = PE + KE MET = Constant п‚§ п‚§ п‚§ п‚§ в–є Note: 1. 2. 3. Maximum kinetic energy is achieved at the lowest point of the pendulum swing. The maximum potential energy is achieved at the top of the swing. When PE is max, KE = 0, and when KE is max, PE = 0. Key Ideas в–є Elastic Potential Energy is the energy stored in a spring or other elastic material. в–є HookeвЂ™s Law: The displacement of a spring from its unstretched position is proportional the force applied. в–є The slope of a force vs. displacement graph is equal to the spring constant. в–є The area under a force vs. displacement graph is equal to the work done to compress or stretch a spring. Key Ideas в–є Springs and pendulums will go through oscillatory motion when displaced from an equilibrium position. в–є The period of oscillation of a simple pendulum is independent of its angle of displacement (small angles) and mass. в–є Conservation of energy: Energy can be converted from one form to another, but it is always conserved.