3.1 Mechanics and Dynamics

Mechanics and Dynamics in Chemical Engineering

Introduction

Mechanics and dynamics form the bedrock of physical sciences, establishing the principles that govern the behavior of bodies subjected to forces. For chemical engineers, these principles are not merely abstract physics; they underwrite the transport of fluids, the design of agitation systems, the operation of particle separation equipment (such as cyclones and centrifuges), and the structural integrity of process piping and vessels. The NCEES FE Chemical exam tests these core concepts through applied vector analysis, particle and rigid body kinematics, friction, momentum, and the principles of work, energy, and power. Success on the exam requires a fluent transition between vector notation and physical systems, alongside rapid application of formulas compiled in the FE Reference Handbook.

Vector Analysis in Engineering Mechanics

Vector operations are essential for resolving forces, moments, and velocities in multi-dimensional space. A vector A is defined in Cartesian coordinates as:

A = A_x * i + A_y * j + A_z * k

where i, j, and k are unit vectors along the x, y, and z axes, respectively. The magnitude of A is given by:

A = sqrt(A_x^2 + A_y^2 + A_z^2)

There are two primary vector products defined in engineering mechanics:

1. Dot Product (Scalar Product): The dot product of two vectors A and B yields a scalar:

A . B = A_x * B_x + A_y * B_y + A_z * B_z = A * B * cos(theta)

where theta is the angle between the two vectors. The dot product is zero if the vectors are orthogonal (theta = 90 degrees). It is primarily used to project a vector along a line of action or to compute work, which is the dot product of force and displacement vectors:

W = F . d

1. Cross Product (Vector Product): The cross product of A and B yields a vector perpendicular to both, following the right-hand rule:

A x B = (A_y * B_z - A_z * B_y) * i - (A_x * B_z - A_z * B_x) * j + (A_x * B_y - A_y * B_x) * k

The magnitude is given by |A x B| = A * B * sin(theta), representing the area of the parallelogram formed by the two vectors. The cross product is non-commutative (A x B = -B x A). Key applications include calculating the moment of a force about a point:

M = r x F

where r is the position vector from the point to the line of action of the force F.

Particle Kinematics and Kinetics

Dynamics is divided into kinematics (the study of motion without regard to forces) and kinetics (the study of forces acting on bodies and the resulting motion). Newton's Second Law is the core governing equation of kinetics:

sum F = m * a

where m is the mass (a measure of translational inertia) and a is the acceleration vector.

For rectilinear kinematics under constant acceleration a, the motion is described by:

1. v = v_0 + a * t
2. s = s_0 + v_0 * t + 0.5 * a * t^2
3. v^2 = v_0^2 + 2 * a * (s - s_0)

where s is position, v is velocity, t is time, and the subscript 0 denotes initial conditions.

In curvilinear motion, particles moving along a curved path of radius r experience centripetal (normal) acceleration directed toward the center of curvature:

a_n = v^2 / r = r * omega^2

where v is the tangential linear velocity, r is the radius of curvature, and omega is the angular velocity (v = r * omega). The tangential acceleration component is:

a_t = dv/dt = r * alpha

where alpha is the angular acceleration. Tangential acceleration represents the rate of change of speed, whereas normal acceleration represents the rate of change of direction.

Friction: Dry and Belt Friction

Friction is the resistive force acting parallel to contact surfaces. NCEES problems focus on two primary types:

1. Dry (Coulomb) Friction: Prior to motion, the static friction force F_f balances the applied tangential forces up to a maximum limit:

F_f,max = mu_s * N

where mu_s is the coefficient of static friction and N is the normal force. Once slipping begins, the friction force drops to a constant kinetic value:

F_f = mu_k * N

where mu_k is the coefficient of kinetic friction (mu_k < mu_s).

1. Belt Friction: When a belt or rope is wrapped around a fixed cylinder, the relationship between the high tension (T_2) and low tension (T_1) is governed by the Capstan Equation:

T_2 = T_1 * e^(mu * beta)

where mu is the coefficient of static (or kinetic) friction, and beta is the total angle of contact in radians. Note that T_2 is always the larger tension (the pulling force), and T_1 is the resisting force.

Momentum, Impulse, and Impacts

Linear momentum (p = m * v) is a vector quantity representing the quantity of motion. The linear impulse-momentum principle, derived by integrating Newton's Second Law over time, states:

I = integral ( F * dt ) = delta p = m * v_2 - m * v_1

If the net external force acting on a system is zero, the total linear momentum is conserved:

sum m_i * v_i,1 = sum m_i * v_i,2

This conservation law is critical for analyzing collisions (impacts). For a direct central impact between two bodies, the coefficient of restitution e measures the elasticity of the collision:

e = (v'_B - v'_A) / (v_A - v_B)

where v_A and v_B are velocities before impact, and v'_A and v'_B are velocities after impact.

• e = 1: Perfectly elastic impact (no kinetic energy loss).
• e = 0: Perfectly plastic (inelastic) impact (bodies stick together, maximizing kinetic energy loss).

Work, Energy, and Power

The concepts of work and energy provide a powerful scalar alternative to solving dynamics problems, bypassing time-integration of forces.

• Work (W) done by a force F along path s is:

W_1->2 = integral ( F . dr )

For a constant force in the direction of displacement, W = F * d. The work done by a linear spring is W = -0.5 * k * (x_2^2 - x_1^2).

• Kinetic Energy (T) represents energy due to motion:
  • Translation: T = 0.5 * m * v^2
  • Rotation (Rigid Body): T_r = 0.5 * I * omega^2 where I is the mass moment of inertia about the axis of rotation.
• Potential Energy (V) represents stored energy:
  • Gravitational: V_g = m * g * h (where h is elevation relative to a datum).
  • Elastic (Spring): V_e = 0.5 * k * x^2 (where x is stretch or compression from unstretched length).

The Work-Energy Principle states that the work done by all external non-conservative forces equals the change in total mechanical energy:

T_1 + V_1 + W_other = T_2 + V_2

If only conservative forces act, mechanical energy is conserved (T_1 + V_1 = T_2 + V_2).

Power (P) is the rate at which work is performed:

P = dW/dt = F . v

For rotational systems, power is the product of torque (tau) and angular velocity (omega):

P = tau * omega

Rigid Body Dynamics and Moment of Inertia

For rigid bodies undergoing rotation, the distribution of mass relative to the axis of rotation is characterized by the mass moment of inertia (I). The parallel axis theorem relates the moment of inertia I about any parallel axis to the moment of inertia I_c about a parallel axis passing through the center of mass:

I = I_c + m * d^2

where d is the perpendicular distance between the two axes. The radius of gyration (r_g) is defined as:

r_g = sqrt(I / m)

Worked Example: Rolling Cylinder on an Incline

A solid cylinder of mass m = 15 kg and radius R = 0.3 m starts from rest and rolls without slipping down a 3.0 m long incline angled at 30 degrees. We must find its linear velocity at the bottom of the incline.

1. Identify the Potential Energy Change: The change in elevation is:

h = d * sin(30 degrees) = 3.0 * sin(30 degrees) = 1.5 m

The potential energy loss is:

delta V = m * g * h = 15 * 9.81 * 1.5 = 220.725 J

1. Express Kinetic Energy for Rolling Without Slipping: A rolling cylinder has both translational and rotational kinetic energy:

T = 0.5 * m * v^2 + 0.5 * I_c * omega^2

For a solid cylinder, I_c = 0.5 * m * R^2. For rolling without slipping, omega = v / R.

T = 0.5 * m * v^2 + 0.5 * (0.5 * m * R^2) * (v / R)^2

T = 0.5 * m * v^2 + 0.25 * m * v^2 = 0.75 * m * v^2

1. Apply Conservation of Energy: Since there is no slipping, static friction does no work, and mechanical energy is conserved:

T_1 + V_1 = T_2 + V_2

0 + m * g * h = 0.75 * m * v^2 + 0

g * h = 0.75 * v^2

v = sqrt( (4 * g * h) / 3 )

v = sqrt( (4 * 9.81 * 1.5) / 3 ) = sqrt( 19.62 ) = 4.43 m/s

The linear velocity of the cylinder at the bottom is 4.43 m/s.