Question

a spring with a hookes law constant of 45 n/m an an unstretched length of 18 cm is hung vertically from a support. a block is attached to the free end of the spring and let go from rest; the lowest point reached by the mass is 30 cm from the support attachment.
determine (a) the amplitude of vibration; (b) the mass of the block, (c) the period and frequency of vibration; and (d) the maximum speed of the block while vibrating.

Explanation:

Step1: Calculate the amplitude

The amplitude \(A\) is half of the total displacement from the highest to the lowest point. The total displacement is \(30 - 18=12\space cm\). So, \(A=\frac{12}{2}=6\space cm = 0.06\space m\)

Step2: Find the mass of the block

At the equilibrium position (mid - point of the motion), the force due to gravity \(mg\) is equal to the spring force \(k\Delta x\). The equilibrium position is at \(18 + 6=24\space cm\) from the support. The extension \(\Delta x=24 - 18 = 6\space cm=0.06\space m\). Using \(mg = k\Delta x\), we can solve for \(m\). \(m=\frac{k\Delta x}{g}\), where \(k = 45\space N/m\) and \(g = 9.8\space m/s^{2}\). So, \(m=\frac{45\times0.06}{9.8}\approx0.276\space kg\)

Step3: Calculate the period and frequency

The period \(T\) of a spring - mass system is given by \(T = 2\pi\sqrt{\frac{m}{k}}\). Substituting \(m = 0.276\space kg\) and \(k = 45\space N/m\), we get \(T=2\pi\sqrt{\frac{0.276}{45}}\approx0.49\space s\). The frequency \(f=\frac{1}{T}\), so \(f=\frac{1}{0.49}\approx2.04\space Hz\)

Step4: Determine the maximum speed

The maximum speed \(v_{max}\) of a simple harmonic oscillator is given by \(v_{max}=A\omega\), where \(\omega=\sqrt{\frac{k}{m}}\). First, \(\omega=\sqrt{\frac{45}{0.276}}\approx12.77\space rad/s\). Then \(v_{max}=0.06\times12.77\approx0.77\space m/s\)

Answer:

(a) The amplitude of vibration is \(0.06\space m\)
(b) The mass of the block is approximately \(0.276\space kg\)
(c) The period is approximately \(0.49\space s\) and the frequency is approximately \(2.04\space Hz\)
(d) The maximum speed of the block while vibrating is approximately \(0.77\space m/s\)