Question

An air-track glider attached to a spring oscillates with a period of 1.50 s . At t=0s the glider is 4.90 cm left of the equilibrium position and moving to the right at 35.6 cm/s .

What is the phase at t=0.5s?

Answer 1

x(t)=Acos(wt+o) x(0)=Acos(o)=-4.90cm

v(t)=-Awsin(wt+o) v(0)=-Awsin(o)=35.6cm/s

A=amplitude w=angular frequency o=phase constant T=period

sin(o)/cos(o) = tan(o) = -v(0)/wx(0)

w= 2(pi)/T = 4(pi)/3

o=arctan(-35.6/(4pi/3)(-4.9) = -1.0478 rad or pi/3

however, since the glider is moving to the right it is not in that quadrant, since tan repeats every pi

subtracting pi will give you the correct answer of -2(pi)/3

o= -2pi/3rad , this is the phase constant

O phase at variable time(in this case 0.5) o= phase constant w=angular frequency t=time

O=wt+o

O=(4pi/3)(0.5) - (2pi/3)

O = 0rad

This works for any other time as well just change the 0.5s to the new time the other two variables won't change.