Angular Impulse in Collision of Two Pivoted Rods | JEE Physics

The sweet spot, or point of percussion, is the optimal point on a hinged rod where applying force produces no hinge reaction force. The universal formula x1 × x2 = kc² relates the distance from hinge to center of mass (x1), distance from center of mass to sweet spot (x2), and radius of gyration about center of mass (kc). For a uniform rod, x2 = l/6. When force is applied closer to the hinge than the sweet spot, the hinge applies force opposite to the applied force; when applied farther, the hinge applies force in the same direction. This explains why hitting a ball at the sweet spot prevents hand injury in sports. For rolling motion, the bottommost point behaves as an instantaneous hinge, and force must be applied at height h = kc²/r from the center of mass to achieve pure rolling without friction.

The center of percussion is the point on a rotating body where an impact produces no reaction force at the pivot. For a uniform rod of length l, the center of percussion is located at a distance x = (2/3)l from the pivot. This is derived from the condition that the impulse produces no moment about the pivot: S × x = 0, which requires x = I/(ma), where I is the moment of inertia about the pivot and a is the distance from the pivot to the center of mass.

The center of mass is the point where the entire mass of an object can be considered to be concentrated for the purpose of analyzing translational motion. The center of percussion is a special point on a rotating body where an impact produces no reaction force at the pivot point. When a force is applied at the center of percussion, the pivot experiences no impulsive reaction, making it the ideal point for striking or applying force without causing unwanted motion at the pivot.

For supported bodies, reaction forces at the pivot must be considered. The center of percussion is the impact point where reaction forces vanish, located at distance h = I_A / (M*h) from the pivot. For a uniform rod pivoted at one end, this is 2/3 the rod length. Practical applications include hammer and tennis racket design, where the grip position should be at the center of percussion to minimize shock transmission to the user's hand.

The center of percussion A is also known as the 'sweet spot' or 'sweet spot' of the rigid body. When a force F is applied at this point, the pivot point experiences no reaction force. This is why it's called the sweet spot - when you hit a baseball bat at this point, you feel no sting in your hands. To prove this, we consider dynamic equilibrium: force equation R - F - mDθ'' = 0, and moment equation about pivot -D*R + P*F - JAθ'' = 0. Solving these equations simultaneously and substituting P = JG/(mD), we find R = 0. This proves that when the force is applied at the center of percussion, the pivot experiences no reaction force.

For two rotating discs colliding inelastically with moments of inertia I1 = 4 and I2 = 2, and angular velocities ω1 = 10 and ω2 = 4, loss in kinetic energy = (1/2)(I1I2/(I1+I2))(ω1-ω2)² = (1/2)(4×2/6)(10-4)² = (4/3)(36) = 48J.

In partially elastic collisions, the kinetic energy after the collision is less than the kinetic energy before the collision. The lost energy is dissipated into other forms such as sound energy (acoustic energy), deformation energy, and thermal energy (heat). This energy transformation occurs because collisions typically involve work of deformation and generate sound.

In perfectly inelastic rotational collisions where bodies stick together, angular momentum is conserved but kinetic energy is lost. Angular momentum conservation: I₁ω₁ + I₂ω₂ = (I₁ + I₂)ω_f. Energy loss: ΔK = (1/2)I₁I₂/(I₁ + I₂)(ω₁ - ω₂)². This shows energy loss depends on the difference in initial angular velocities and the moments of inertia.

This segment covers collision dynamics including elastic and inelastic collisions. In elastic collision, bodies collide and separate without sticking. Two conservation laws apply: Conservation of Momentum (m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂) and Conservation of Kinetic Energy. These equations show relative velocity of approach equals relative velocity of separation: u₁ - u₂ = v₂ - v₁. The coefficient of restitution e = (v₂ - v₁)/(u₁ - u₂) measures collision 'bounciness' and equals 1 for perfectly elastic collisions. In inelastic collision, bodies stick together. Conservation of momentum gives m₁u₁ + m₂u₂ = (m₁ + m₂)v. Energy loss is calculated as ΔKE = [(1/2)m₁u₁² + (1/2)m₂u₂²] - [(1/2)(m₁ + m₂)v²]. This energy is converted to heat, sound, or deformation. The segment emphasizes that in inelastic collisions, kinetic energy is not conserved, unlike elastic collisions.

For elastic collisions, relative velocity before equals relative velocity after (u1 - u2 = v2 - v1), derived from momentum and kinetic energy conservation. For completely inelastic collisions, bodies stick together with common velocity v = (m1u1 + m2u2)/(m1 + m2). Kinetic energy loss = (1/2)(m1m2/(m1 + m2))(u1 - u2)², showing maximum energy loss when bodies stick together.

Impact problems can be unified using matrix notation: M·v⁺ + J^T·λ = M·v⁻ + S - e·J^T·μ, where M is the mass matrix, v⁺/v⁻ are velocities after/before impact, J is the Jacobian of constraints, λ contains Lagrange multipliers (impulses), S represents external applied impulses, and μ represents impulsive constraints. This framework handles both joint constraints and contact impacts simultaneously, extending Newton's impact law to complex multibody systems.

For a two-degree-of-freedom planar manipulator with links l1 and l2, centroids at distances lc1 and lc2 from joints, the complete dynamics analysis proceeds through forward and backward recursions. Forward recursion yields ω1 = q̇1 k, α1 = q̈1 k, ω2 = (q̇1 + q̇2) k, and α2 = (q̈1 + q̈2) k. Centroid accelerations include centripetal and tangential components. Gravity projects onto local axes as Fgx = -m_i g cos(q_total), Fgy = -m_i g sin(q_total). Backward recursion calculates reaction forces and joint torques, revealing that even small torques produce significant oscillations in joint forces (up to 700N and 4000N) due to centripetal and Coriolis effects, demonstrating why dynamic loading analysis is essential for joint design.

This segment covers the theoretical foundation of multi-body dynamics. Newton's second law gives the centroidal equations: m × r_double_dot = F (translational) and J × phi_double_dot = N (rotational). These can be expressed in matrix form as M × c_double_dot = H. When kinematic joints are present, reaction forces must be included. Newton's third law states that reaction forces act between bodies along axes where no relative motion is allowed. A pin joint removes two translational degrees of freedom, resulting in two orthogonal reaction force components (lambda_1 and lambda_2) acting on each body, with equal magnitudes but opposite directions.

Forces of constraint (reaction forces) enforce geometric restrictions on particle motion. These forces are unknown a priori and must be determined alongside particle trajectories. As systems become more complex (double pendulums, rigid bodies), the number of constraints and reaction forces increases. Each particle may experience multiple reaction forces from connected constraints. This complexity makes analytical solutions increasingly difficult, requiring advanced techniques beyond simple Newtonian approaches.

This section demonstrates analyzing multi-link mechanisms in equilibrium. For each hinge, include reaction forces (X and Y components) and any applied moments. To find reactions at internal hinges, cut the mechanism at the hinge and consider one part, replacing the removed part with reaction forces. Write moment equilibrium equations about the hinge point to eliminate unknown forces. The moment arm is the perpendicular distance from the reference point to the force line of action, calculated using trigonometric functions. The direction of rotation determines the sign convention.