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): $$T = 2\pi\sqrt{\frac{m}{k}}$$
  $$= 2\pi\sqrt{\frac{0.1 \texttt{ kg}}{100 \texttt{ N/m}}}$$ $$= 0.628 \texttt{ s}$$

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

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

Solution:

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

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

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): $$A = 10 \texttt{ cm} = 0.1 \texttt{ m}$$

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

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

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): $$T = 2\pi\sqrt{\frac{m}{k}}$$
  $$= 2\pi\sqrt{\frac{0.5 \texttt{ kg}}{100 \texttt{ N/m}}}$$ $$= 1.13 \texttt{ s}$$

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: $$T_{10} = 16 \texttt{ s}$$

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

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

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): $$A = 5 \texttt{ cm} = 0.05 \texttt{ m}$$

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

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