The displacement of a spring vibrating in damped harmonic motion is given by y(t) = 4e^-3t sin(2pi*t) where y = displacement and t = time with t greater than/equal to zero. Find the time(s) when the spring is at its equilibrium position (y=0). The number "e" is Euler's number.
I have absolutely no clue what this is even asking. I have written down the information minus the extra words but I am completely lost. Please help?
I am not sure what the exponent is
the way you typed it using no brackets, it would be
(4e^-3) * tsin(2πt)
or is it
4e^( -3tsin(2πt) )
sorry
y(t) =4e(then the exponent -3t) all times sin(2pi*t)
y = 4^(-3t) * sin(2πt)
so if we set y = 0
4^(-3t) = 0 ----> so solution
or
sin(2πt) = 0
2πt = 0 or 2πt = π or 2πt = 2π or ..
t = 0 or t = 1/2 or t = 1 or t= ..
looks like every half second, starting with t = 0
Of course, I can help you with that! To find the time(s) when the spring is at its equilibrium position (y=0), we need to solve the equation y(t) = 0.
Let's substitute y(t) with 0 in the given equation:
0 = 4e^(-3t)sin(2πt)
Now, we need to solve for t. To do that, we can divide both sides of the equation by 4:
0 = e^(-3t)sin(2πt)
Now, we have two factors on the right-hand side: e^(-3t) and sin(2πt).
Since the product is zero, one or both of the factors must be equal to zero.
For e^(-3t) = 0, we know that e is Euler's number, which is approximately equal to 2.71828. But e raised to any power can never be zero, so this factor cannot be zero.
Now let's consider sin(2πt) = 0.
In order to find the values of t for which sin(2πt) = 0, we look for the values of t for which the sine function is equal to zero. The sine function is equal to zero at multiples of π, so we have:
2πt = nπ
Where n is an integer. To find t, we can divide both sides of the equation by 2π:
t = n/2
Therefore, the values of t for which the spring is at its equilibrium position (y=0) are given by t = n/2, where n is an integer.
To summarize, the time(s) when the spring is at its equilibrium position (y=0) are given by t = n/2, where n is an integer.