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

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 \( a_{\text{max}} = 0.8 , \text{m/s}^2 \)