JEE Advanced:

1. A spring-mass system with a mass of 0.1 kg and a spring constant of 100 N/m is set into oscillation with an amplitude of 0.1 m.

Solution:

• Period (T): $$T=2\pi \sqrt{\frac{m}{k}}$$
  $$=2\pi \sqrt{\frac{0.1\mathtt{\text{ kg}}}{100\mathtt{\text{ N/m}}}}$$ $$=0.628\mathtt{\text{ s}}$$

• Frequency (f): $$f=\frac{1}{T}$$ $$=\frac{1}{0.628\mathtt{\text{ s}}}$$ $$=1.59\mathtt{\text{ Hz}}$$

2. A simple pendulum with a length of 1 m is set into oscillation with an amplitude of 10 degrees.

Solution:

• Period (T): $$T=2\pi \sqrt{\frac{L}{g}}$$
  $$=2\pi \sqrt{\frac{1\mathtt{\text{ m}}}{9.81{\mathtt{\text{ m/s}}}^2}}$$ $$=2.01\mathtt{\text{ s}}$$

• Frequency (f): $$f=\frac{1}{T}$$ $$=\frac{1}{2.01\mathtt{\text{ s}}}$$ $$=0.498\mathtt{\text{ Hz}}$$

3. A block of mass 2 kg is attached to a spring of spring constant 500 N/m. The block is displaced from its equilibrium position by 10 cm and released. Find the amplitude, period, and frequency of oscillation.

Solution:

• Amplitude (A): $$A=10\mathtt{\text{ cm}}=0.1\mathtt{\text{ m}}$$

• Period (T): $$T=2\pi \sqrt{\frac{m}{k}}$$
  $$=2\pi \sqrt{\frac{2\mathtt{\text{ kg}}}{500\mathtt{\text{ N/m}}}}$$ $$=0.632\mathtt{\text{ s}}$$

• Frequency (f): $$f=\frac{1}{T}$$ $$=\frac{1}{0.632\mathtt{\text{ s}}}$$ $$=1.58\mathtt{\text{ Hz}}$$

CBSE Board:

1. A mass of 0.5 kg is attached to a spring with a spring constant of 100 N/m. The mass is set into oscillation with an amplitude of 0.2 m. Calculate the period of oscillation.

Solution:

• Period (T): $$T=2\pi \sqrt{\frac{m}{k}}$$
  $$=2\pi \sqrt{\frac{0.5\mathtt{\text{ kg}}}{100\mathtt{\text{ N/m}}}}$$ $$=1.13\mathtt{\text{ s}}$$

2. A simple pendulum with a length of 0.8 m is set into oscillation. If the time taken for 10 oscillations is 16 seconds, calculate the amplitude of oscillation.

Solution:

• Period (T) for 10 oscillations: $$T_{10}=16\mathtt{\text{ s}}$$

• Period (T) for 1 oscillation: $$T=\frac{T_{10}}{10}$$ $$=\frac{16\mathtt{\text{ s}}}{10}$$ $$=1.6\mathtt{\text{ s}}$$

• Amplitude (A): $$T=2\pi \sqrt{\frac{L}{g}}$$ $$1.6\mathtt{\text{ s}}=2\pi \sqrt{\frac{0.8\mathtt{\text{ m}}}{g}}$$ $$A=0.16\mathtt{\text{ m}}$$

3. A mass of 1 kg is attached to a spring of spring constant 200 N/m. The block is displaced from its equilibrium position by 5 cm and released. Determine the amplitude, period, and frequency of oscillation.

Solution:

• Amplitude (A): $$A=5\mathtt{\text{ cm}}=0.05\mathtt{\text{ m}}$$

• Period (T): $$T=2\pi \sqrt{\frac{m}{k}}$$
  $$=2\pi \sqrt{\frac{1\mathtt{\text{ kg}}}{200\mathtt{\text{ N/m}}}}$$ $$=0.444\mathtt{\text{ s}}$$

• Frequency (f): $$f=\frac{1}{T}$$ $$=\frac{1}{0.444\mathtt{\text{ s}}}$$ $$=2.25\mathtt{\text{ Hz}}$$