JEE-Main

1. Normal force = Weight of block - Force applied perpendicular to the surface

$$N = mg - F\sin\theta$$

$$N = (1\text{ kg})(9.81\text{ m/s}^2) - (10\text{ N})(\sin 30\degree)$$

$$N = \boxed{8.54\text{ N}}$$

2. Maximum height reached by a projectile:

$$h=\frac{v_i^2\sin^2\theta}{2g}$$

Where:

• (h) is the maximum height reached (in meters)
• (v_i) is the initial velocity (in m/s)
• (\theta) is the angle of projection (in degrees)
• (g) is the acceleration due to gravity (9.81 m/s^2)

Substituting the given values:

$$h=\frac{(10\text{ m/s})^2\sin^290\degree}{2(9.81\text{ m/s}^2)}$$

$$h = \boxed{4.58\text{ m}}$$

3. Work done by a spring:

$$W=\frac{1}{2}kx^2$$

Where:

• (W) is the work done (in Joules)
• (k) is the spring constant (in N/m)
• (x) is the displacement of the spring (in meters)

Substituting the given values:

$$W=\frac{1}{2}(100\text{ N/m})(0.1\text{ m})^2$$

$$W = \boxed{0.5\text{ J}}$$

4. Gravitational force acting on a satellite:

$$F_g=\frac{Gm_1m_2}{r^2}$$

Where:

• (F_g) is the gravitational force (in Newtons)
• (G) is the gravitational constant (6.674 × 10-11 N·m^2/kg^2)
• (m_1) is the mass of the earth (5.972 × 10^24 kg)
• (m_2) is the mass of the satellite (100 kg)
• (r) is the distance between the center of the earth and the satellite (6.371 × 10^6 m + 400 × 10^3 m)

Substituting the given values:

$$F_g=\frac{(6.674 × 10^{-11}\text{ N·m}^2/\text{kg}^2)(5.972 × 10^{24}\text{ kg})(100\text{ kg})}{(6.371 × 10^6\text{ m} + 400 × 10^3\text{ m})^2}$$

$$F_g = \boxed{5.93\times10^2 \text{ N}}$$

CBSE-Boards

1. Normal force acting on a book:

$$N = mg$$

Where:

• (N) is the normal force (in Newtons)
• (m) is the mass of the book (in kilograms)
• (g) is the acceleration due to gravity (9.81 m/s^2)

Substituting the given values:

$$N = (1\text{ kg})(9.81\text{ m/s}^2)$$

$$N = \boxed{9.81\text{ N}}$$

2. Maximum height reached by a projectile:

$$h=\frac{v_i^2\sin^2\theta}{2g}$$

Where:

• (h) is the maximum height reached (in meters)
• (v_i) is the initial velocity (in m/s)
• (\theta) is the angle of projection (in degrees)
• (g) is the acceleration due to gravity (9.81 m/s^2)

Substituting the given values:

$$h=\frac{(5\text{ m/s})^2\sin^290\degree}{2(9.81\text{ m/s}^2)}$$

$$h = \boxed{1.27\text{ m}}$$

3. Work done by a spring:

$$W=\frac{1}{2}kx^2$$

Where:

• (W) is the work done (in Joules)
• (k) is the spring constant (in N/m)
• (x) is the displacement of the spring (in meters)

Substituting the given values:

$$W=\frac{1}{2}(50\text{ N/m})(0.05\text{ m})^2$$

$$W = \boxed{0.0625\text{ J}}$$

4. Gravitational force acting on a satellite:

$$F_g=\frac{Gm_1m_2}{r^2}$$

Where:

• (F_g) is the gravitational force (in Newtons)
• (G) is the gravitational constant (6.674 × 10-11 N·m^2/kg^2)
• (m_1) is the mass of the earth (5.972 × 10^24 kg)
• (m_2) is the mass of the satellite (50 kg)
• (r) is the distance between the center of the earth and the satellite (6.371 × 10^6 m + 200 × 10^3 m)

Substituting the given values:

$$F_g=\frac{(6.674 × 10^{-11}\text{ N·m}^2/\text{kg}^2)(5.972 × 10^{24}\text{ kg})(50\text{ kg})}{(6.371 × 10^6\text{ m} + 200 × 10^3\text{ m})^2}$$

$$F_g = \boxed{2.96\times10^2 \text{ N}}$$