Shortcut Methods
Typical Numerical Problems on Simple Harmonic Motion For JEE Exam:
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Problem: A mass of 0.5 kg is attached to a spring with a spring constant of 100 N/m. Find the period and frequency of oscillation. Solution:
- Period (T) = $$2\pi\sqrt{\frac{0.5 kg}{100 N/m}} = \boxed{0.63 s}$$
- Frequency (f) = $$\frac{1}{0.63 s} = \boxed{1.59 Hz}$$
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Problem: A spring-mass system has a mass of 2 kg and a spring constant of 200 N/m. If the system is displaced from its equilibrium position by 10 cm and released, find the amplitude and energy of oscillation. Solution:
- Amplitude (A) = $$\frac{0.1 m}{\sqrt{2}} = \boxed{0.071 m}$$
- Energy (E) = $$\frac{1}{2}\cdot 200 N/m\cdot (0.071 m)^2 = \boxed{0.504 J}$$
For CBSE Board Exam:
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Problem: A simple pendulum has a length of 1 m. What is the time period of oscillation of the pendulum? Solution:
- Time Period (T) = $$2\pi\sqrt{\frac{1 m}{9.81 m/s^2}} = \boxed{2.01 s}$$
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Problem: A mass of 0.2 kg is attached to a vertical spring with a spring constant of 50 N/m. If the mass is pulled down 5 cm from its equilibrium position and released, find the amplitude and frequency of oscillation. Solution:
- Amplitude (A) = $$0.05 m = \boxed{0.05 m}$$
- Frequency (f) = $$\frac{1}{2\pi}\sqrt{\frac{50 N/m}{0.2 kg}} = \boxed{1.13 Hz}$$
