Question

A body of mass 200g is suspended from the end of a spiral spring whose force constant is 0.8N/m. The body is set into a simple harmonic motion with amplitude 20cm. Calculate
A) the period of the motion
B) the frequency of the motion
C) the angular speed
D) the total energy
E) the maximum velocity of the motion
F) maximum acceleration

Answer 1

Let's solve each part of the problem step by step using the relevant formulas for simple harmonic motion.

**Given:**
- Mass \( m = 200 \, \text{g} = 0.2 \, \text{kg} \)
- Spring constant \( k = 0.8 \, \text{N/m} \)
- Amplitude \( A = 20 \, \text{cm} = 0.2 \, \text{m} \)

### A) Period of the motion (\( T \))
The formula for the period \( T \) of a simple harmonic oscillator is given by:
\[
T = 2\pi \sqrt{\frac{m}{k}}
\]
Substituting the values:
\[
T = 2\pi \sqrt{\frac{0.2 \, \text{kg}}{0.8 \, \text{N/m}}} = 2\pi \sqrt{\frac{0.2}{0.8}} = 2\pi \sqrt{0.25} = 2\pi \cdot 0.5 = \pi \approx 3.14 \, \text{s}
\]

### B) Frequency of the motion (\( f \))
The frequency \( f \) is the reciprocal of the period:
\[
f = \frac{1}{T}
\]
Calculating:
\[
f = \frac{1}{\pi} \approx \frac{1}{3.14} \approx 0.318 \, \text{Hz}
\]

### C) Angular speed (\( \omega \))
The angular speed \( \omega \) is related to the frequency by the formula:
\[
\omega = 2\pi f
\]
Calculating:
\[
\omega = 2\pi \cdot 0.318 \approx 2.0 \, \text{rad/s}
\]

### D) Total energy (\( E \))
The total energy in simple harmonic motion can be calculated using the formula:
\[
E = \frac{1}{2} k A^2
\]
Calculating:
\[
E = \frac{1}{2} \cdot 0.8 \cdot (0.2)^2 = \frac{1}{2} \cdot 0.8 \cdot 0.04 = 0.016 \, \text{J}
\]

### E) Maximum velocity (\( v_{\text{max}} \))
The maximum velocity in simple harmonic motion is given by:
\[
v_{\text{max}} = A \omega
\]
Calculating:
\[
v_{\text{max}} = 0.2 \cdot 2.0 = 0.4 \, \text{m/s}
\]

### F) Maximum acceleration (\( a_{\text{max}} \))
The maximum acceleration in simple harmonic motion is given by:
\[
a_{\text{max}} = A \omega^2
\]
Calculating:
\[
a_{\text{max}} = 0.2 \cdot (2.0)^2 = 0.2 \cdot 4 = 0.8 \, \text{m/s}^2
\]

### Summary of Results:
- A) Period \( T \approx 3.14 \, \text{s} \)
- B) Frequency \( f \approx 0.318 \, \text{Hz} \)
- C) Angular speed \( \omega \approx 2.0 \, \text{rad/s} \)
- D) Total energy \( E \approx 0.016 \, \text{J} \)
- E) Maximum velocity \( v_{\text{max}} = 0.4 \, \text{m/s} \)
- F) Maximum acceleration \( a_{\text{max}} = 0.8 \, \text{m/s}^2 \)