Work, Energy, Impulse, Momentum, Rotation, and Vibration

Choose an integral method

Newton's second law is the base model for dynamics, but FE Mechanical problems often move faster with work-energy or impulse-momentum. Use work-energy when a force, spring, weight, or torque acts through a displacement and the question asks for speed, height, compression, or stopping distance. Use impulse-momentum when a force acts over time, a collision occurs, or the question asks for velocity before and after impact.

The advantage is that these methods avoid solving acceleration at every instant. The risk is leaving out a force that does work or an impulse. Normal forces do no work if displacement is tangent to the surface, but friction usually removes energy. Internal collision forces cancel in a system momentum balance, but external impulses do not.

Work and energy setup

Write the energy states before formulas. For particles and translating bodies, include kinetic energy T = mv^2/2, gravitational potential V_g = mgh, spring potential V_s = kx^2/2, and nonconservative work such as friction. Pick a datum and keep h signs consistent. If friction acts over distance s, its work is negative with magnitude f_k s.

Power is work rate. For translation, P = Fv when force and velocity are collinear. For rotation, P = T omega. Convert rpm to rad/s with omega = rpm(2pi)/60 before using rotational power or kinetic energy.

Momentum and collisions

Linear momentum is mv. Impulse is integral F dt and equals change in momentum. If the net external impulse is negligible in a direction, momentum is conserved in that direction. Energy is not automatically conserved in collisions. Perfectly elastic collisions conserve kinetic energy; perfectly plastic collisions, where bodies stick together, conserve momentum but lose kinetic energy.

For angular impulse-momentum about a fixed axis, use angular momentum H = I omega and angular impulse = integral torque dt. Make sure the mass moment of inertia I is about the same axis used for omega and torque.

Rotation model checks

For fixed-axis rotation, angular kinematics mirrors linear constant-acceleration equations when alpha is constant. Use theta in radians, not revolutions, unless the formula explicitly uses cycles. Tangential acceleration is alpha r and normal acceleration is omega^2 r. A point on a rotating disk can have both components.

Vibration essentials

Single-degree-of-freedom vibration questions usually test recognition. For an undamped spring-mass system, natural circular frequency is omega_n = sqrt(k/m). Frequency in hertz is f_n = omega_n/(2pi). For a torsional system, use the torsional stiffness and mass moment of inertia analog.

Damping ratio tells the response shape: underdamped if zeta < 1, critically damped at zeta = 1, and overdamped if zeta > 1. Forced vibration and isolation questions often focus on resonance and transmissibility. If forcing frequency is near natural frequency and damping is low, expect amplification. If isolation is the goal, the operating frequency generally must be well above natural frequency.