To determine the maximum displacement of the tuning fork, we start by examining the given equation for displacement:
[tex]\[ d = 0.4 \sin (1760 \pi t) \][/tex]
The sine function, \(\sin(x)\), oscillates between -1 and 1. In other words:
[tex]\[ -1 \leq \sin (1760 \pi t) \leq 1 \][/tex]
This means that the value of \(\sin(1760 \pi t)\) ranges from -1 to 1. To find the maximum displacement, we need to consider the maximum value that \(\sin(1760 \pi t)\) can attain, which is 1.
Let’s substitute this maximum value into the displacement equation:
[tex]\[
d_{\text{max}} = 0.4 \sin(1760 \pi t) \quad \text{where} \quad \sin(1760 \pi t) = 1
\][/tex]
This gives us:
[tex]\[
d_{\text{max}} = 0.4 \times 1 = 0.4 \text{ mm}
\][/tex]
Thus, the maximum displacement of the tuning fork is:
[tex]\[ \boxed{0.4 \text{ mm}} \][/tex]
