Define Gravitational Force and state its property. State and drive Newton's law of Universal Gravitation. Explain Gravitational Field and derive the expression for Gravitational Potential and Intensity.

Part 1a: Moment of inertia

Step 1: Recall the formula for the moment of inertia of point masses rotating about an axis.

$I = m_1 r_1^2 + m_2 r_2^2$

Step 2: Convert distances to meters and substitute the given values ($m_1 = 2\ \mathrm{kg}$, $r_1 = 10\ cm = 0.10\ \mathrm{m}$, $m_2 = 3\ \mathrm{kg}$, $r_2 = 20\ cm = 0.20\ \mathrm{m}$).

$r_1^2 = (0.10\ m)^2 = 0.010\ \mathrm{m^2}$

$m_1 r_1^2 = 2\ kg \times 0.010\ m^2 = 0.020\ \mathrm{kg \cdot m^2}$

$r_2^2 = (0.20\ m)^2 = 0.040\ \mathrm{m^2}$

$m_2 r_2^2 = 3\ kg \times 0.040\ m^2 = 0.12\ \mathrm{kg \cdot m^2}$

Step 3: Add the contributions.

$I = 0.020\ kg \cdot m^2 + 0.12\ kg \cdot m^2 = 0.14\ \mathrm{kg \cdot m^2}$

0.14 \mathrm{kg \cdot m^2}

Part 1b: Angular acceleration

Step 1: Recall the relation between torque, moment of inertia, and angular acceleration.

$\tau = I \alpha$

$\alpha = \frac{\tau}{I}$

Step 2: Substitute the given torque $\tau = 5\ \mathrm{N \cdot m}$ and $I = 0.14\ \mathrm{kg \cdot m^2}$.

Express $I$ as a fraction: $0.14 = \frac{7}{50}$.

$\alpha = \frac{5\ N \cdot m}{\frac{7}{50}\ kg \cdot m^2} = 5\ N \cdot m \times \frac{50}{7\ kg \cdot m^2} = \frac{250}{7}\ \mathrm{rad/s^2}$

Step 3: Approximate numerical value.

$\frac{250}{7} \approx 35.7\ \mathrm{rad/s^2}$

\dfrac{250{7} rad/s^2}