The motion of a wave traveling along a stretched string is described by the equation:
y(x,t)=(2.4cm)sin(5.7x−14.0t)
What is the minimum time it takes for a particle on the string to move from
y= -2.4 cm to y = 2.4 cm?
Can someone help me give some hints on how to do the question
half a period
14 t changes by pi
say from 14t = 0 to 14t = pi
or
t = pi/14
Sure! To find the minimum time it takes for a particle on the string to move from y = -2.4 cm to y = 2.4 cm, we need to find the time when the sine function within the equation reaches its maximum value.
Let's break down the equation: y(x,t) = (2.4 cm) sin(5.7x - 14.0t)
The amplitude of the wave is given as 2.4 cm, and it is multiplied by a sine function that depends on two variables: x and t.
To find the time at which the particle reaches its maximum displacement, we need to look at the argument of the sine function, which is (5.7x - 14.0t). We want this argument to be equal to its maximum value, which is 90 degrees or π/2 radians.
Since the sine function has a period of 2π, we can set up an equation to solve for t:
5.7x - 14.0t = π/2
Now, we can solve for t:
14.0t = 5.7x - π/2
t = (5.7x - π/2) / 14.0
To find the minimum time, we need to consider the maximum possible value of x. Since the sine function has a maximum displacement of 2.4 cm, the maximum value of x that would allow the particle to reach this displacement is when the argument of the sine function is equal to π/2:
5.7x = π/2
x = π/2 / 5.7
Now, substituting this value of x back into the equation for t:
t = (5.7 * (π/2 / 5.7) - π/2) / 14.0
Simplifying this expression will give you the minimum time it takes for the particle on the string to move from y = -2.4 cm to y = 2.4 cm.