A string vibrates according to the equation
y' = (0.30 cm) sin[(π/3 cm-1)x] cos[(55π s-1)t]
What is the speed of a particle of the string at the position x = 1.5 cm when t = 9/8 s?
This is a standing wave.
y at constant x = 1.5
=.3 sin [1.5(pi/3-1)] cos t(55 pi -1)
so
dy/dt at constant x = 1.5
={.3 sin [1.5(pi/3-1)]} * {(1-55pi)sin [(9/8)55pi-1)] }
I kinda wrote my equation wrong so heres a new version:
y' = (0.30 cm) sin[(π/3 cm^(-1))x] cos[(55πs^(-1))t]
I thought it was messed up. The method I gave you is how to go about it anyway.
To find the speed of a particle on a string at a specific position and time, we need to find the first derivative of the equation with respect to time (t).
Given the equation: y' = (0.30 cm) sin[(π/3 cm^-1)x] cos[(55π s^-1)t]
To find the first derivative with respect to time, we only need to differentiate the term cos[(55π s^-1)t], as the other term does not depend on time.
Differentiating cos[(55π s^-1)t] with respect to t gives: -55π sin[(55π s^-1)t].
Now that we have the first derivative, we can substitute the values of x = 1.5 cm and t = 9/8 s into the equation to find the speed of the particle at that specific position and time.
Substituting x = 1.5 cm and t = 9/8 s, we get: -55π sin[(55π s^-1)(9/8 s)].
Calculating this value will give us the speed of the particle at the given position and time.