Lecture 3 Examples: Simple Harmonic Motion A spring is mounted horizontally, wit

Question

Lecture 3 Examples: Simple Harmonic Motion A spring is mounted horizontally, with its left end fixed (see figure). A spring balance attached to the free end and pulled toward the right indicates that the stretching force is proportional to the displacement and a 6.0 N force causes a displacement of 0.030 m We replace the spring balance with a 0.50-kg glider, pull it 0.020 m to the right along a frictionless air track, and release it from rest. Find the force constant k of the spring. Find the angular frequency Omega. frequency f, and period T of the resulting oscillation. If we give the same glider an initial displacement x0 = + 0.015m and an initial velocity of Vn = +0.40 m/s Find the period, amplitude and phase angle of the resulting motion. Write equations for the displacement, velocity and acceleration as functions of time.

Explanation / Answer

1a)k=F/d=6/0.03=200N/m

b)w=sqrt(k/m)=sqrt(200/0.5)=20rad/s

f=w/2pi=3.183Hz

T=1/f=0.314s

2 a)period is the same. Let the amplitude be A.

Conserving energy,

0.5mv0x^2+0.5kx0^2=0.5kA^2

0.5*0.5*0.4^2+0.5*200*0.015^2=0.5*200*A^2

A=0.025m

Initial phase=cos^-1(0.015/0.025)=53.1 degrees or 0.9273 radians

b)x=0.025Cos(20t+0.9273)

v=dx/dt=-0.5Sin(20t+0.9273)

a=dv/dt=-10Cos(20t+0.9273)

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