Download Simple Harmonic Motion - General Physics I - Lecture Slides and more Slides Physics in PDF only on Docsity! Simple Harmonic Motion docsity.com Force of a Stretched Spring • If a spring is pulled to extend beyond its natural length by a distance x, it will pull back with a force where k is called the “spring constant”. The same linear force is also generated when the spring is compressed. • A Natural length F kx= − Extension x F kx= − Spring’s force docsity.com Energy in SHM: Potential Energy Stored in the Spring • Plotting a graph of external force F = kx as a function of x, the work to stretch the spring from x to x + Δx is force x distance • ΔW = kxΔx, so the total work to stretch the spring to x0 is • A 0 21 02 0 x W kxdx kx= =∫ x x0 F 0 kx0 Δx kx This work is stored in the spring as potential energy. docsity.com Potential Energy U(x) Stored in Spring • The potential energy curve is a parabola, its steepness determined by the spring constant k. • For a mass m oscillating on the spring, with displacement the potential energy is • X U(x) U(x) = ½kx2 x0( )cosx A tω φ= + ( ) ( )2 212 cosU x kA tω φ= + docsity.com Total Energy E for a SHO • The total energy E of a mass m oscillating on a spring having constant k is the sum of the mass’s kinetic energy and the spring’s potential energy: • E = ½mv2 + ½kx2 • For a given E, the mass will oscillate between the points x = A and –A, where • E = ½kA2 • Maximum speed is at x = 0, where U(x) =0, and E = ½mv2 at x = 0 • X E = K + U U(x) = ½kx2 x0 A-A docsity.com F = ma for the Simple Pendulum • The displacement along the circular arc is x = ℓθ. • The restoring force is F = -mgsinθ ≅ -mgθ = -mgx/ℓ along the arc. • F = ma is d2x/dt2 = −gx/ℓ (canceling out m from both sides!). • v ℓ θ mgsinθ m docsity.com Period of the Simple Pendulum • The equation of motion has solution • Here and the time for a complete swing • v ℓ θ mgsinθ m ( )cosx A tω φ= + /gω = 2 / 2 / .T gπ ω π= = 2 2/ /d x dt gx= − The time for a complete swing doesn’t depend on the mass m, for the same reason that different masses fall at the same rate. docsity.com Reminder: the Conical Pendulum • Imagine a conical pendulum in steady circular motion with small angle θ. • As viewed from above, it moves in a circle, the centripetal force being . • So the equation of motion is and for the x-component of • v 2 2/ /d x dt gx= − ℓ θ m Top View: r ( )/mg r− ( )2 2/ /d r dt g r= − r r docsity.com
