PHYSICS 1051 WORKSHOP 2: SIMPLE HARMONIC MOTION
Problems involving phase and the phase constant
1. What is the connection between phase constant φ and the time-dependence of x(t)?
x
x(t) = A cos(ωt+0)
A
x(0)=A
v(0)=0 (slope of x vs t at t=0)
t
-A
T= 2π/ω
x
A
x(t) = A cos(ωt+φ)
x(0)=A cos φ
t
-A
v(0)= -Aω sin φ
T= 2π/ω
Time where cos(ωt+φ) = +1 so (ωt+φ) = 0 and t = - φ/ω
For this graph, is φ positive or negative? Why?
2. Finding x(t)=A cos(ωt+φ) from initial conditions.
-Need: (a) enough information about system (i.e. T or k and m) to
get ω
(b) two more pieces of information (could be x(0) and v(0))
-Example in class: used x(0) = A cos φ, v(0) = -Aω sin φ, and T to
get
ω = 2π/ T,
A  x0  v0 /  
2
2
, and   tan 1  v0 /x0 
Must be careful with quadrant for φ!
2. Finding x(t)=A cos(ωt+φ) from initial conditions (continued).
-Can have other combinations of given information:
EXAMPLE: given x(0), v(0), and A, find parameters in
x(t) = A cos(ωt+φ)
SUGGESTION:
Start from x(t) = A cos(ωt+φ) and v(t) = -Aω sin(ωt+φ)
Apply initial conditions (x(0), v(0)). Do they fix unknowns?
x(0) = A cos φ → φ = cos-1(x(0)/A) → two possible values for φ
v(0) = -Aω sin φ → select φ to get correct sign for v(0)
Then ω = - v(0)/(A sin φ) → must be positive!
3. Could be asked to find time corresponding to specific condition in motion.
EXAMPLE:
For motion described by x(t) = A cos(ωt+φ) where A =7 cm, ω = 5
rad/s, and φ = -π/6, find the earliest positive time for which vx(t) = +20
cm/s.
FIRST:
Note that vx(t) = -Aω sin(ωt+φ) = -35 cm/s sin (5t - π/6)
SO:
Need to solve sin (5t - π/6) = -20/35 = -0.571
Careful: multiple solutions!
SOLVE: sin (5t - π/6) = -20/35 = -0.571
sin θ
θ
sin θ = -.571
θ = -0.608 rad
θ = π+0.608
rad
sin-1(-0.571) = - 0.608 rad
or
sin-1(-0.571) = π + 0.608 rad
or
sin-1(-0.571) = 2π - 0.608 rad
etc.
For sin-1(-0.571) = (5t - π/6) = - 0.608 rad , get t = -0.17 s
→ NOT POSITIVE
For sin-1(-0.571) = (5t - π/6) = π + 0.608 rad , get t = +0.855 s
→ POSITIVE
Second solution makes sense:
T= 2π/ω = 1.26 s so that 0.855s is a little larger than T/2.