Typical Numericals on Problem-Solving: Simple Harmonic Motion JEE Main Exam:

1. A mass-spring system:

• a) Amplitude: $$0.1\text{ m}$$
• b) Angular frequency: $$10\text{ rad/s}$$
• c) Maximum velocity: $$0.1\text{ m/s}$$

2. A simple pendulum:

• a) Time period: $$2\pi \sqrt{\frac{1}{9.81}}\approx 2.01\text{ s}$$
• b) Maximum angular velocity: $${\omega }_{max}=\sqrt{\frac{g}{l}}\sin {\theta }_0=\sqrt{\frac{9.81}{1}}\sin {10}^{\circ }\approx 0.174\text{ rad/s}$$
• c) Maximum potential energy: $$U_{max}=mgl(1-\cos {\theta }_0)=1\times 9.81\times 1(1-\cos {10}^{\circ })\approx 0.173\text{ J}$$

3. Particle in SHM:

• a) Maximum velocity: $$2\pi fA=2\pi \times 2\times 0.2=2.51\text{ m/s}$$
• b) Maximum acceleration: $${\omega }^{2}A=(2\pi f{)}^{2}A=(2\pi \times 2{)}^2\times 0.2=31.42{\text{ m/s}}^2$$
• c) Total energy: $$E=\frac{1}{2}k{A}^2=\frac{1}{2}m{\omega }^2{A}^2=3.14\text{ J}$$

4. Block-spring system:

• a) Time period: $$T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{2}{10}}\approx 2.83\text{ s}$$
• b) Angular frequency: $$\omega =\sqrt{\frac{k}{m}}=\sqrt{\frac{10}{2}}=2.24\text{ rad/s}$$
• c) Maximum velocity: $$\omega A=2.24\times 0.5=1.12\text{ m/s}$$

CBSE Board Exam:

1. Mass-spring system:

• a) Amplitude: $$0.2\text{ m}$$
• b) Time period: $$2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{1}{100}}\approx 0.63\text{ s}$$
• c) Maximum velocity: $$A\omega =0.2\times \sqrt{\frac{100}{1}}=2\text{ m/s}$$

2. Simple pendulum:

• a) Time period: $$2\pi \sqrt{\frac{l}{g}}=2\pi \sqrt{\frac{2}{9.81}}\approx 2.83\text{ s}$$
• b) Maximum angular velocity: $${\omega }_{max}=\sqrt{\frac{g}{l}}\sin {\theta }_0=\sqrt{\frac{9.81}{2}}\sin {15}^{\circ }\approx 0.54\text{ rad/s}$$
• c) Maximum potential energy: $$U_{max}=mgl(1-\cos {\theta }_0)=1\times 9.81\times 2(1-\cos {15}^{\circ })\approx 1.07\text{ J}$$

3. Particle in SHM:

• a) Maximum velocity: $$2\pi fA=2\pi \times 5\times 0.3=9.42\text{ m/s}$$
• b) Maximum acceleration: $${\omega }^{2}A=(2\pi f{)}^{2}A=(2\pi \times 5{)}^2\times 0.3=188.50{\text{ m/s}}^2$$
• c) Total mechanical energy: $$E=\frac{1}{2}K{A}^2=\frac{1}{2}m{\omega }^2{A}^2=13.88\text{ J}$$

4. Block-spring system:

• a) Amplitude: $$0.6\text{ m}$$
• b) Time period: $$T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{4}{20}}\approx 1.88\text{ s}$$
• c) Maximum velocity: $$\omega A=2\sqrt{5}\times 0.6\approx 1.69\text{ m/s}$$

Note: These are just sample calculations for better understanding. Numerical values may vary based on the actual problem statements and given parameters.