Displacement x(t) is a sine function of time
x(t) = A sin (2 π f t) (when the body is at t= 0 s in a positive direction through the central position)
- x(t) displacement at t = t (s)
- A amplitude (x en A : the same unit of length)
Example
A body is in a simple harmonic motion. f = 20.0 Hz. A = 5.0 cm
At t= 0 s is the body in a positive direction through the central position . In other words φ(0) = 0
Find the displacement at t = 0.018 s
x = A sin (2 π f t) = 5.0 sin (2 π x 20.0 x 0.018) = 5.0 sin (2.26) = 5.0 x 0,77 = 3.85 cm
Remark: set calculator to RADIANS !!!
Example
At what times (t < 0.05 s) in the above example is the displacement x = +3.0 cm ?
x(t) = A sin (2 π f t)
3.0 = 5.0 sin (2 π 20 t)
3.0/5.0 = sin (2 π 20 t)
0.6 = sin (2 π 20 t) ( sin-1 0.6 = 0.64 )
0.64 = 2 π 20 t
t = 5.12 x 10-3 s
The second time :
t= 0.025 – 5.12 x 10-3 = 0.0198 s =
19.8 x 10-3 s