Dynamics

Two halves: kinematics and kinetics

Dynamics is the analysis of bodies whose velocity is changing. It is conventionally divided into two domains, and keeping them straight is the single most useful organizing habit in the subject. Kinematics studies the motion of objects in space and time without considering the forces that cause it: position, velocity, acceleration, and the geometric constraints that link one body's motion to another's. Kinetics studies motion in relation to mass and force. The two connect through the equation of motion. Kinetics produces that equation by relating forces to accelerations; kinematics takes the equation, together with the constraint equations and the initial conditions, and turns it into an actual trajectory. The link runs both ways: if you know the forces and the initial conditions you can predict the motion, and if you can measure the motion you can back out the forces.

This matters because a question that looks hard is often only kinematic. "How long does the arm take to swing 90 degrees at this angular acceleration?" needs no force at all. Recognizing that saves you from drawing a diagram you did not need.

Rectilinear motion, curvilinear motion, and the projectile trick

Rectilinear motion is motion along a straight line, and one axis is enough to describe it. Curvilinear motion follows a curved path and needs at least two. The bridge between them is that motions along perpendicular axes are independent: what happens along x does not constrain what happens along y. Projectile motion is the cleanest use of that fact. Once the object is in flight, with air resistance neglected, the only force on it is its weight, so the horizontal acceleration is zero and the vertical acceleration is g downward. The horizontal problem is constant-velocity motion; the vertical problem is constant-acceleration motion. They share only the clock. Two one-dimensional problems, solved separately, then recombined.

For launch and landing at the same height, that analysis yields the range R = v0^2 sin(2*theta0)/g, which is largest at 45 degrees and gives equal ranges for complementary angles such as 15 and 75 degrees.

Name the idealization out loud, because it is doing real work. Neglecting air resistance is exactly what makes the horizontal acceleration zero. Drag depends on air density, the square of the velocity, viscosity, compressibility, and the body's size, shape and inclination to the flow; engineers collect all of that into D = Cd * rho * V^2 * A / 2 with a drag coefficient measured in a wind tunnel. Because drag scales with the square of speed, the drag-free model is decent for a thrown ball over a short arc and useless for a golf ball, an arrow, or an artillery shell. A projectile answer is only as good as the sentence that says drag was ignored.

Normal and tangential components: curvature is acceleration

When a path curves, it is often easier to attach the coordinate axes to the moving particle than to the ground. In normal-tangential coordinates, one unit vector points along the direction of travel and the other points inward, perpendicular to it, toward the center of curvature. The acceleration then splits cleanly:

a = (dv/dt) * u_t + (v^2 / rho) * u_n

The tangential component is the rate of change of speed. The normal component depends only on speed and rho, the radius of curvature of the path at that instant. The two are perpendicular, so the total acceleration is their vector sum, not their arithmetic sum.

Read that equation slowly, because it contains the correction most students need. Setting dv/dt to zero — constant speed — does not zero the acceleration. It leaves v^2/rho, pointed inward. Velocity is a vector, and changing its direction is a change in velocity just as much as changing its magnitude. A car holding 25 m/s around a bend, a satellite in a circular orbit, and a point on the rim of a flywheel spinning at constant rate are all accelerating continuously. Only a straight path at unchanging speed has zero acceleration, because only then are both components zero.

Circular motion, the centripetal requirement, and the centrifugal mistake

Circular motion is the special case where rho is constant and equal to the circle's radius r. In uniform circular motion the acceleration is purely normal, with magnitude a_c = v^2/r, directed at the center. Combining that with the second law gives the centripetal force requirement F_c = m v^2 / r.

Centripetal force is not a new kind of force, and it never belongs on a free-body diagram as a separate arrow. It is a requirement that the real forces already present must satisfy. For a ball on a string, tension supplies it. For a car on a flat road, friction does. For a satellite, gravity does. For a car on a banked curve, the horizontal component of the normal force does; the ideal banking angle satisfies tan(theta) = v^2/(r*g), which contains no mass, so the same bank works for a motorcycle and a loaded truck. If the available force cannot meet the requirement, the body does not travel in the circle — it leaves the path.

That brings us to centrifugal force. In an inertial frame there is no outward force on the body. There is one net inward force, and the body's own inertia, which is not a force. The outward push a passenger feels is the seat and door supplying the inward force to the passenger's body while the passenger's inertia resists being turned. If you deliberately write the equations in a frame rotating with the car, a centrifugal term does appear — along with a Coriolis term — but these are fictitious inertial forces, bookkeeping introduced by the non-inertial frame, with no physical source and no third-law partner. In the inertial frame where you will do nearly all your work, they are simply absent.

Relative motion and reference frames

Every velocity is a velocity with respect to something. State the frame or the number is meaningless. Velocities compose by vector addition: the velocity of a particle P relative to frame S equals its velocity relative to an intermediate frame S' plus the velocity of S' relative to S, written v_PS = v_PS' + v_S'S, with the shared subscript adjacent on the right-hand side so the intermediate frame cancels visually.

One consequence deserves emphasis. If S' moves at constant velocity relative to S, then the relative acceleration is zero and both observers measure the same acceleration for P. That is why the second law holds unchanged in any frame moving at constant velocity relative to an inertial one, and why the accelerating or rotating frames are the ones that force fictitious terms into the equations.

Rigid bodies turning about a fixed axis

A rigid body is not a point, so describing where it is takes more than a position vector. For rotation about a fixed axis, the extra variable is angular position theta, related to arc length by theta = s/r and measured in radians. Its derivatives are angular velocity omega = d(theta)/dt in rad/s and angular acceleration alpha = d(omega)/dt in rad/s^2. Every point on the body shares the same theta, omega, and alpha; what differs is r, so a point at radius r has tangential speed v = omegar and tangential acceleration alphar, while carrying a centripetal component omega^2*r as well.

The kinetics needs a rotational stand-in for mass. That is the moment of inertia, I = sum of m_j * r_j^2, in kgm^2. It is the quantitative measure of rotational inertia exactly as mass is the measure of linear inertia, and the r^2 is the whole story: mass far from the axis counts far more than the same mass near it. That is why a hollow cylinder is harder to spin up than a solid one of the same mass and outer radius, and why I is not a property of the body alone but of the body together with the chosen axis. A uniform thin disk about its central axis has I = (1/2)mR^2, and the parallel-axis theorem shifts a known result to a parallel axis a distance d away as I = I_cm + md^2.

With that, the rotational analogue of the second law is sum of M = Ialpha. In engineering practice the rigid-body pair is written together — sum F = ma for the center of mass and sum M = I*alpha about the axis — and the standing rule is that the moments and the moment of inertia must be taken about the same point, either the axis of rotation O or the center of mass G. When the center of mass sits on the axis the rotation is balanced, its acceleration is zero, and the force equations reduce to the bearing reactions. When it does not, the center of mass accelerates and the bearings carry that too, which is why rotating machinery is balanced on purpose.

Three routes, and choosing between them

Engineering dynamics offers three method families for both particles and rigid bodies, and they answer different questions from the same physics. The direct force-acceleration route applies sum F = ma and sum M = Ialpha at an instant; use it when you want an acceleration, a bearing reaction, or a required torque right now. The work-energy route integrates force over distance and connects speeds at two positions without ever computing acceleration; use it when a problem gives you positions and speeds and does not ask about time. The impulse-momentum route integrates force over time and connects velocities at two instants; use it for impacts, for short bursts of large force, and for anything where several bodies interact and the interaction forces are not worth chasing individually. Work and energy, and impulse and momentum, are developed in their own lessons in this unit; the point here is that they are alternatives to the equation you just learned, not additions to it.

A workable order of questions: is this kinematic only? If not, does the question mention time — impulse-momentum — or distance and speed — work-energy? If neither, or if you need a force at a specific instant, go direct. Pick the route that avoids computing the quantity you do not care about.

A closing note on scope. These lessons are educational material, not engineering design guidance. Real analysis of a machine, vehicle, rotor, or structure requires a licensed engineer working to the governing code, with verified properties, measured loads, and margins chosen for the consequences of failure. Every idealization used above — rigid body, no drag, constant g, frictionless pivot, fixed axis — is an assumption you would have to defend in practice.