Question

An air-track glider attached to a spring oscillates with a period of 2.66 s. At t=0 s the glider is 2.1 cm left of the equilibrium position and moving to the right at 39.62 cm/s.
a)What is the phase constant? (in degrees)

b) What is the phase at t=0 s (in degrees)

c) What is the phase at t=0.5s (in degrees)

d) What is the phase at t=1 s? (in degrees)

e) What is the phase at t=1.5 s? (in degrees)

Answer 1

x(t)=Acos(wt+o) x(0)=Acos(o)=-5.10cm
v(t)=-Awsin(wt+o) v(0)=-Awsin(o)=37.0cm/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

a) o=arctan(-39.62/(4pi/3)(-2.1) = 1.047179915rad 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

O phase at variable time(in this case 0.5) o= phase constant w=angular frequency t=time
b)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. Hope you understood and good luck.