We can solve this problem using the principles of simple harmonic motion (SHM). Here's the given data:

• Amplitude $A$ = $8 \, \text{mm} = 0.008 \, \text{m}$
• Frequency $f$ = $120 \, \text{Hz}$
• Angular frequency $\omega = 2 \pi f = 2 \pi \cdot 120 \, \text{rad/s} = 240 \pi \, \text{rad/s}$
• Time $t$ = $0.5 \, \text{s}$

The general equation for SHM is: $x(t) = A \cos(\omega t + \phi)$ Assuming no phase shift ($\phi = 0$):

---

(i) Displacement at $t = 0.5 \, \text{s}$

Substitute $A = 0.008 \, \text{m}$, $\omega = 240 \pi$, and $t = 0.5 \, \text{s}$: $x(0.5) = 0.008 \cos(240 \pi \cdot 0.5)$

Simplify $240 \pi \cdot 0.5 = 120 \pi$, which is an integer multiple of $2 \pi$: $\cos(120 \pi) = \cos(60 \cdot 2\pi) = \cos(0) = 1$ Thus: $x(0.5) = 0.008 \, \text{m} = 8 \, \text{mm}$

---

(ii) Maximum blade speed

The maximum speed occurs when the blade passes through the equilibrium point, and is given by: $v_{\text{max}} = \omega A$ Substitute $\omega = 240 \pi$ and $A = 0.008$: $v_{\text{max}} = 240 \pi \cdot 0.008 = 1.92 \pi \, \text{m/s}$ Numerically: $v_{\text{max}} \approx 6.03 \, \text{m/s}$

---

(iii) Magnitude of the maximum acceleration of the blade

The maximum acceleration occurs at the extreme positions and is given by: $a_{\text{max}} = \omega^2 A$ Substitute $\omega = 240 \pi$ and $A = 0.008$: $a_{\text{max}} = (240 \pi)^2 \cdot 0.008 = 57600 \pi^2 \cdot 0.008$ $a_{\text{max}} = 4608 \pi^2 \, \text{m/s}^2$ Numerically: $a_{\text{max}} \approx 45453.2 \, \text{m/s}^2$

---

Final Results:

1. Displacement at $t = 0.5 \, \text{s}$: $8 \, \text{mm}$
2. Maximum blade speed: $\approx 6.03 \, \text{m/s}$
3. Magnitude of maximum acceleration: $\approx 45453.2 \, \text{m/s}^2$

Would you like detailed derivations or explanations of any specific steps?

---

Related Questions:

1. How is the displacement in SHM derived?
2. Why is the maximum speed proportional to $\omega A$?
3. How does the angular frequency affect the motion in SHM?
4. Can the maximum acceleration be used to determine $A$ if unknown?
5. What is the effect of doubling the frequency on $v_{\text{max}}$ and $a_{\text{max}}$?

---

Tip:

In SHM problems, always determine $\omega = 2\pi f$ first, as it governs both speed and acceleration.