Free-body — the series · Illustrated Physics

A block on a 27° incline. Free-body diagram of a block resting on an inclined plane: weight straight down, resolved into a component pressing into the surface and one pulling along it, with friction opposing the slide.

The slope, and the two directions that make it easy

An inclined plane looks like a harder problem than a flat one. Split the weight into two components chosen to suit the slope and it becomes an easier one.

part 1 · mechanics

Which failure comes first. The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a paperback, standing, a full filing cabinet, a pint glass.

Slide or topple

Push a wardrobe and it goes over; push a brick and it skids. Both are held by the same friction and both are pushed by the same hand, and which of the two failures arrives first has nothing to do with how hard the push is. The floor decides it, by shifting where it pushes back.

part 2 · mechanics

Four supports, and a whole line of answers. The same top on four supports at the corners of a square, with the load in the same place. Five sets of reactions are drawn and every one of them satisfies all three equilibrium equations exactly — worst residual 1.1e-16 across the whole family — so statics does not prefer any of them. They differ by a multiple of the pattern plus, minus, plus, minus around the square: pressing one diagonal pair harder and the other pair less adds no net force and no moment about either axis, which is exactly what it means for the problem to have a fourth unknown and only three equations. The rigid-body idealisation has not been applied carelessly here; it has been applied correctly, and the answer it gives is that there is no answer. Every member drawn keeps all four reactions positive, so the requirement that a leg can only push narrows the family without closing it. What decides is left out of the model entirely — how much each leg gives under load.

The table statics cannot settle

A rigid top on three legs has one possible set of reactions and a rigid top on four has infinitely many, all of them balancing every force and every moment exactly. The extra leg does not make the problem harder; it makes it unanswerable, and the answer has to come from somewhere the model deliberately threw away.

part 3 · mechanics

The stress a temperature change puts into a bar that cannot move. The stress in a member held between supports that will not let it change length, against how much its temperature changes, for four materials. Each line is EαΔT, computed here from a free expansion and the force needed to undo it, and checked by evaluating it for a bar half a metre long and one thirty-seven metres long: the two agree to every figure carried, because neither the length nor the cross-section appears in the answer. steel develops 2.40 MPa for every kelvin and reaches yield at 104 K; aluminium develops 1.59 MPa for every kelvin and reaches yield at 151 K; concrete develops 0.30 MPa for every kelvin and reaches cracking at 10 K; invar develops 0.17 MPa for every kelvin and reaches yield at 1655 K. Concrete reaches its cracking stress after ten kelvin, which is less than a sunny afternoon, and is why every slab has movement joints in it. Invar is in the comparison because it was made to have a small product: it is as stiff as steel and develops a fourteenth of the stress, which is a statement about the expansion coefficient and nothing else.

The load nobody applied

A redundant structure develops forces with nothing on it. Change its temperature and the same extra constraint that made statics unanswerable also refuses the expansion — and a restrained steel member reaches its yield stress after a hundred and four kelvin, a figure that contains no length, no area and no load.

part 4 · mechanics

Four stress states in one beam, and only one of them has no tension. Stress across the depth of a 300 by 600 millimetre concrete section at the middle of an eight-metre span, compression to the right, for four conditions. Under the load alone the bottom fibre is in tension at 11.1 MPa, which is four times what concrete can carry, so an ordinary reinforced beam cracks there and relies on steel to hold the crack together. With 1,500 kilonewtons of prestress 120 millimetres below the centroid and the full load applied, the section runs from 9.8 to 3.6 MPa and every fibre of it is in compression. The second case is the one that surprises: with the prestress applied and nothing whatever to oppose it, the top fibre is in tension at 1.7 MPa, because a force below the centroid bends the beam upwards. A prestressed beam is at its most vulnerable when nothing is on it, and what rescues it is its own weight: adding that alone brings the top back to 0.3 MPa of compression. Each stress block is checked by integrating it and recovering the force and the moment that produced it.

A state no load could reach

The free direction a redundant structure leaves open can be driven on purpose. Tighten a tendon through a concrete beam and its whole stress state moves into the half of the range the material is good at; tighten a bolt hard and the load it carries fluctuates by a fifth of what is applied to it. Both put the structure somewhere no arrangement of external loads could.

part 5 · mechanics

The one number the tolerances cannot touch

A redundant structure's load sharing depends on stiffnesses and manufacturing errors that nobody knows. Its collapse load does not depend on either. Once members yield they hold a known force instead of a force proportional to a displacement, the compatibility equations that needed the unknowns drop out, and the load at which the structure becomes a mechanism follows from a work balance with no stiffness in it at all.

part 6 · mechanics

Where the thread's line meets the floor. A spool resting on the floor on its rims of radius R, its thread wound round a hub of radius r = 0.5R and leaving the hub's underside tangentially, pulled up and to the right at 20°, 60° and 78° above the horizontal; the dashed lines continue each thread's line back down to the floor. The spool turns about the contact point, and the sense of the turn is set by where the thread's line meets the floor. At 20° it meets the floor behind the contact point, on the far side from the hand, and the spool rolls towards the hand. At 60°, where cos θ = r/R, it passes exactly through the contact point; the pull has no turning effect there, and the spool stays still while the hand pulls. At 78° it meets the floor in front of the contact point, between the spool and the hand, and the spool rolls away from the hand, paying the thread out.

The spool that rolls towards the hand pulling it

Pull gently on the thread of a cotton reel lying on its side on a table, with the thread coming off the underside of the hub, and the reel rolls towards the hand, winding the thread up as it comes. Raise the thread and at one particular angle the reel will not move at all however hard it is pulled; raise it further and the reel rolls away, paying the thread out. Which way it goes is decided by a single line — where the thread's direction, extended, meets the table — and the same line decides which way a bicycle goes when its lower pedal is pulled backwards.

part 7 · mechanics

Eight blocks leaning past the edge of a table. 8 identical blocks of length 1 stacked on a table edge (the vertical line), each set as far out as balance allows: the dot on each interface marks the centre of mass of all the blocks above it, which sits exactly on the edge of the block beneath. The top block overhangs the second by 1/2, the second the third by 1/4, the third the fourth by 1/6, and the k-th by 1/(2k); together the stack reaches 1.359 block lengths beyond the table — the top block lies wholly beyond it, since 1.359 is more than one. Every interface is balanced on its edge, so the stack stands, but with no margin at all.

The stack that leans past its own table

Stack identical books on the edge of a table, each pushed out as far as it will go without falling, and four of them are enough for the top book to lie entirely beyond the table's edge. Keep going and the overhang grows without limit — the stack can reach as far out as anyone likes — but only as half the harmonic series, so that two book-lengths need 31 books, three need 227 and five need more than twelve thousand. The arithmetic is the same that settles whether a ladder leaning on a wall slips or a wardrobe topples, and the slow growth is the price of having nothing but the stack's own weight to hold it back.

part 8 · mechanics

The bending a falling column must carry, along its height. The bending moment at each height of a uniform column toppling about its base, in units of its weight times its length, at tilts of 10°, 30°, 60° from upright. Every curve peaks at a third of the height, at mgL·sinθ/27: 0.0064, 0.0185, 0.0321. The base and the top carry none. The same column held still at the same tilt would carry its greatest moment at the base, mgL·sinθ/2 — 0.250 at 30°, 13.5 times as much. Falling relieves the bending by a factor of 13.5 and moves it a third of the way up.

The chimney that breaks a third of the way up

A tall brick chimney felled by demolition rarely reaches the ground in one piece. It snaps partway up, early in its fall, and the top falls behind the rest. Nothing has hit it. The cause is a free-body diagram: as the column rotates about its base, the material near the top must be dragged round faster than its own weight would take it, and the material lower down holds back. The bending this needs is largest a third of the way up, at a thirteenth and a half of what the same column would carry standing still at the same tilt — and brick, which carries no tension, cracks there after only a few degrees of lean.

part 9 · mechanics