Shortcut Methods and Tricks for Numerical Problems

Chapter: Motion of System of Particles and Rigid Bodies

1. Momentum (p) of a particle:

• Formula: $$p = mv$$
• Shortcut: Simply multiply the mass (m) by the velocity (v) of the particle.

2. Kinetic energy (K) of a car:

• Formula: $$K = (1/2)mv^2$$
• Shortcut: Multiply half of the mass (m) by the square of the velocity (v) of the car.

3. Impulse (J) applied to a body:

• Formula: $$J = F \times t$$
• Shortcut: Multiply the force (F) by the time duration (t) during which the force is applied.

4. Final velocity (v) of a body from rest:

• Formula: $$v = u + (F/m) \times t $$
• Shortcut: Add the initial velocity (u, which is 0 in this case) to the product of force (F) per unit mass (m) and the time (t) of force application.

5. Final velocity (v) of a moving body:

• Formula: $$v = u + (F/m) \times t$$
• Shortcut: Add the initial velocity (u) to the product of acceleration due to the force (F/m) and the time (t) over which the force acts.

6. Final velocity (v) of bodies after collision (assuming perfect inelastic collision)

• Formula: $$V_{final} = [(m_1u_1 + m_2u_2)/(m_1 + m_2)]$$
• Shortcut: Divide the sum of the products of masses (m) and initial velocities (u) of both bodies by the total mass of the combined bodies.

7. Angular momentum (L) of a rotating body:

• Formula: $$L = I\omega$$
• Shortcut: Multiply the moment of inertia (I) of the body by its angular velocity (ω).

8. Angular impulse (J) applied to a body:

• Formula: $$\tau = I\omega$$
• Shortcut: Multiply the torque (τ) acting on the body by the time (t) during which the torque is applied.

9. Final angular velocity (ω) of a rotating body with applied torque:

• Formula: $$\omega_{final} = \omega_{initial} + (\tau/I) \times t$$
• Shortcut: Add the initial angular velocity (ω) to the product of acceleration due to torque (τ/I) and the time (t).

10. Final angular velocity (ω) of multiple colliding bodies:

• Formula: $$\omega_{common} = [(I_1\omega_1 + I_2\omega_2 + …)/(I_1 + I_2 + …)]$$
• Shortcut: Divide the sum of the products of moments of inertia (I) and initial angular velocities (ω) of all bodies by the total moment of inertia of the system.

These shortcut methods and tricks provide efficient and simplified approaches to solving numerical problems related to the motion of systems of particles and rigid bodies.