Phy 202: General Physics II

Phy 202: General Physics II Ch 10: Simple Harmonic Motion & Elasticity

The Ideal Spring & Hooke’s Law • Springs are objects that exhibit elastic behavior • An ideal spring is: • Massless (the mass of the spring is negligible compared to • The applied force (Fapplied) required to compress/stretch is proportional to the displacement of the spring from its unstrained length (x) or Fapplied = kx where k is called the spring constant (or stiffness of the spring) • To stretch/compress a spring, the spring exerts a restoring force of equal & opposite magnitude (reaction force, F) against the stretching/compressing force or F = -kx {this is referred to as Hooke’s Law!}

The (Elastic) Restoring Force (& Newton’s 3rd Law) Action: • Applied force is proportional to displacement of the spring: Fapplied = kx Reaction: • Restoring force is equal/opposite to applied force: F = -Fapplied = -kx

Work Performed by an Ideal Spring • When an ideal spring is stretched a displacement x by an applied force, the average force applied to the spring is Favg = ½ kx • The work performed to stretch the spring is W = (Favgcos q)x = ½ kx2 • The strained spring therefore gains an elastic potential energy PEelastic = ½ kx2

Displacing an Ideal Spring results in Simple Harmonic Motion • Releasing a strained a spring results in oscillating motion due to the spring restoring force • Motion oscillates between (+x and –x) • This type of motion is called Simple Harmonic Motion

Newton’s 2nd Law & Ideal Springs • Applying Newton’s 2nd Law to a stretched ideal spring: SF = ma = -kx The acceleration of the spring is a = - (k/m).x • The acceleration of the spring at any point in the motion is proportional to the displacement of the spring • For motions of this type, the angular frequency (w) of the motion is w =(k/m)½ • The period of the motion (T) is T = 1/f = 2p/w = 2p.(m/k)½ General form: a = w2 x (when a ~ x)

Simple Harmonic Motion • When the restoring force of a spring obeys Hooke’s Law (F=-kx), the resulting motion is called Simple Harmonic Motion • Consider a mass attached to a stretched spring that is released at to=0: • The displacement (x) of the mass due to the spring’s restoring force will be x = A cos wt where A is the amplitude of the strained spring • The velocity (v) of the mass will be v = -Aw sin wt = -vmax sin wt where vmax = Aw • The acceleration (a) of the mass will be a = -Aw2 cos wt = -amax cos wt where amax = Aw2

Graphical Perspective of SHO Displacement (x) x = Acos(wt) Velocity (v) v = -vmaxsin(wt) Acceleration (a) a = -amaxcos(wt)

Conservation of Energy & Simple Harmonic Motion • When work is performed on a spring due to stretch/compression by an applied force the spring gains potential energy equal to PEelastic = ½ kx2 • As the spring is released and the restoring force w/in the spring drives the motion of the spring(assuming no friction) • PEelastic is converted to KE as the spring force does work • When the spring’s length equals its unstretched length, all of the PEelastic is converted to KE • Applying conservation of Energy to the spring: (PEelastic)stretched = KEunstretched or ½ kx2 = ½ mv2 • Therefore, the speed of the spring at its unstretched length is related to the length of the original displacement of the spring: v = (k/m)½

The Pendulum • Consider the motion of a mass (m) attached to a string (length, l): • The gravitational force (mg) exerts a torque on the mass (at all but the bottom point of the swing) t = Ia = mgl sinq since I = ml2 and sin q ~ q a = (mgl/I) sinq = (mgl/ml2) sinq a = (g/l).sinq or a = (g/l).q • The period of the motion (T) is therefore T = 2p(l/g)½ Note: a ~ q similar to a ~x (for a spring) therefore this motion looks like the form: a = w2q q l m mg

Elastic Deformation Types of deformation: • Stretching/compression Fstretch/compress = Y(DL /Lo)A • Shear deformation Fshear = S (DX/Lo)A • Volume deformation DP = -B (DV/Vo)

Stress, Strain & Hooke’s Law • We can consider a strained mass as though it were a collection of small masses attached by a system of springs • Since deformation is related to the applied force: Fstretch/compress = (YA/Lo )DL • The effective spring constant (keffective) for the mass is • keffective = YA/Lo • Note that keffective is inversely proportional to Lo • Question: Consider a long spring (spring constant=k). How do the spring constants of the smaller pieces (k’) compare to the original k?

Phy 202: General Physics II