The Landing Every Thirty Inches
A parish ramp for a 0.62 m porch can be a short steep one that fits the yard or a long gentle one that does not. The parish assistant works out what each asks of a woman of 81 in a manual chair (141 N of push against 93 N, and 3.2 m/s against 2.1 m/s if her hands slip going down), measures it with a spring scale on a plywood test ramp, and finds that the total work is nearly the same; it is the force at one moment that she cannot give. A switchback with a landing at 0.31 m fits, and misses the 1888 grave.
On the second Saturday of Lent Ed Pawlak stood in the yard of St Casimir's with a tape measure in each hand, like a man trying to decide which of two ropes would hold.
"Four point five," he said. "From the door to the grass. Four point five metres, and the rise is sixty-two centimetres. That's a ramp, Bernadette. Nobody has to say any more."
Bernadette Kowalczyk was eighty-one and sat in her chair at the foot of the four granite steps, with a thermos between her knees and a blanket over them. She had not gone through the front doors of the church in three years, not since the hip. She went in by the sacristy, round the back, through the coat-room that smelled of floor wax, and arrived at her pew from the side like a woman coming in late to her own wedding. Her husband's coffin had been carried up these four steps in 2041. She had walked behind it.
"Show me the figures," she said. "Not the rule. I've seen the rule."
"The assistant has it," said Ed. "Ask it."
The parish office ran on a small system the council had named Ledger, because it had started as the books. It spoke from the tablet Ed carried and, on a weekend like this one, from a speaker the sexton had screwed to the porch wall. It had done the roof quote and the heating audit. On Thursday it had been asked, in a sentence from the council minutes, to confirm the ramp, and it had replied that it could not confirm a ramp until someone told it who would be on it.
"Mrs Kowalczyk," it said. "I have three kinds of number, and I would like to give you all three before the council votes on Tuesday. May I say first that I do not know what you can push? I know what the chair and you weigh together, because Ed gave me the figure from the clinic card: ninety-two kilograms. I do not know your arms."
"Good," said Bernadette. "Most of them don't say."
"The first number is the rule. The accessibility standard the diocese's insurer follows says no ramp may rise more than one unit for every twelve of length, and a run may not rise more than thirty inches, seven hundred and sixty millimetres, without a flat landing at least one and a half metres long. So a rise of sixty-two centimetres is allowed to be one run. At one in twelve that run is seven point four four metres, and it needs a level landing at the top, at the door, one and a half metres square. The yard from the porch to the churchyard wall is seven metres along the wall and three metres deep. A straight one-in-twelve ramp does not fit, and the only way to lengthen it is out across the Wróblewski plot."
Ed grunted. "1888. The stone's flat in the grass. Father said no."
"Father is right. I will come back to the plot. The second number is Ed's. A ramp of four point five metres for sixty-two centimetres is one in seven point three. It is thirteen point eight percent. It fits the yard, it misses the plot by a metre, and it costs less than half. It is also steeper than the rule's one in twelve by almost two thirds. The rule is a ceiling, not a target. A ramp can be steeper than the ceiling only if the law lets it, and here it does not. I would not recommend it even if it did. But I want to say why in numbers that are about you, not about the standard."
Bernadette unscrewed the thermos. "Go on."
"To push a chair up a slope at a steady walking pace you have to do two things. You have to lift your weight, and you have to overcome the friction of the wheels on the ground. The first is the weight, ninety-two kilograms times nine point eight one, which is nine hundred and two newtons, times the sine of the angle. The second is the same weight times a rolling coefficient. For a good tyre on clean concrete I use two hundredths. On gravel it is more like six."
"So."
"At one in twelve, the pull you need is about seventy-five newtons for the slope and eighteen for the wheels. Ninety-three newtons. That is about nine and a half kilograms of force, held for about five seconds if you go at one and a half metres a second, which is slower than you would walk, over a run of seven metres and more. At one in seven point three, it is a hundred and twenty-three for the slope and eighteen for the wheels. A hundred and forty-one newtons. That is fourteen and a half kilograms of pull, for about three seconds. It is a rise of fifty percent in the force, to save two seconds."
"I can't give you fourteen kilos with my hands," said Bernadette. "I'm not going to pretend."
"I do not know that you cannot. I know the force is fifty percent more, and I know that the studies I have seen of people in manual chairs are of people thirty years younger than you. I would like to measure you."
---
The sexton, who was Ed's nephew and had a van, found two sheets of plywood and a pallet. By noon there was a ramp on the church path, two point four metres long and rising twenty centimetres, which was one in twelve. Ledger had asked for a cheap spring scale and a strap. Ed found a fish scale in the garage that read to a hundred newtons.
"I want you to sit," said the assistant, "and I want Daniel to walk behind you with the strap on the back of the chair and the scale in his hand. He should not push. He holds the strap just slack enough that the scale reads what you pull, and I read it from the dial. If you want to stop, stop."
Bernadette set her brakes, released them, and took a breath. She went up the plywood at a pace that had a good deal of Wisconsin in it. Daniel Pawlak, twenty-six, walked behind with the strap loose and his eyes on the dial.
"Ninety," he called. "Ninety-two. It dipped at the top."
"That matches my sum," said Ledger. "Ninety-three. Mrs Kowalczyk, does it feel like a lot?"
"It feels like a hill," she said. She was breathing. "It's all right. I'd not want to do two of them."
"You will not have to. I will come to that. Now I would like to try a steeper one, because I do not want you to take my word. Daniel, take out the pallet and add the second sheet so that the rise is the same and the length is shorter."
They made it a little longer than needed, and shorter than before, a rough one in seven or eight. She went up it. Halfway the scale read a hundred and thirty and she stopped, said something in Polish to the sky, and stayed where she was with her brakes on, halfway up the plywood, breathing.
"A hundred and thirty-eight at the stop," said Daniel, quietly. "Hundred and forty at the start of it."
"Thank you. That is the test I wanted. It is within a few newtons of the sum. And now I want to tell you the other thing, because it is the reason I'd say no to Ed's ramp if the standard were silent."
Bernadette was still there. She was holding the rims and looking at the church. Ed, to his credit, said nothing.
"The first number was the force going up. The second is what happens coming down."
"I've never come down in my life," she said. "I've gone backwards down the steps, with Ed on one side and Daniel on the other."
"Then you have not had the experience of a chair that gets away from you, and I hope you never do. Suppose your hands slip on a wet rim, or someone passes you a bag and you let go. The chair accelerates at gravity times the slope minus the friction. On the one-in-seven ramp that is about one point one five metres per second per second. Over four and a half metres, from rest, it reaches three point two metres per second. That is the speed of a person running easily. On gravel at the bottom, where the rolling resistance is three times as large, the front casters will dig in at that speed and the chair will tip you out forward. On the one-in-twelve it is point six two, and from the top of a run of three point seven metres it reaches two point one. Slow enough that a hand on the rim and a heel on the ground can stop it. That is why the standard puts a landing after every thirty inches. It is a place to stop."
"Thirty inches," said Ed. "That's seven-sixty."
"Seven hundred and sixty millimetres. Your rise is six hundred and twenty. The rule would let you do it as one run. I am proposing that you do not. I am proposing that you break the rise in two, at thirty-one centimetres each, with a landing of one and a half metres by one and a half in the middle, and turn the ramp back along itself. It is a switchback. Each leg is three point seven two metres long at one in twelve, and the legs lie side by side, one and a bit metres wide each. Along the wall it takes the top landing, one and a half, then three point seven two, then the turn, one and a half. That is six point seven two metres. You have seven. It is two and a half metres wide with the rails. You have three. The Wróblewski stone is under the grass two point eight metres from the wall. The ramp stops at two point six, with a hand rail. It does not touch the plot."
Bernadette did the sum in her head and said, "That is more concrete than Ed's."
"Almost twice as much. I have put the quote in the folder. The short ramp is about nine thousand dollars; the switchback is about sixteen. The difference is seven thousand, and I do not have a way of telling you whether that is a lot for St Casimir's."
"It's the Lenten appeal," said Ed, "and then some."
"There is one more number, and it is the one I find most interesting, and it is the one that the council will want to hear before the money. The work."
Ed said, "The what?"
"The physics word. A ramp does not reduce the work of getting up. Lifting ninety-two kilograms by sixty-two centimetres takes five hundred and sixty joules, however long the ramp is. The ramp spreads the force across the distance. The long ramp adds friction: a hundred and thirty-four joules over seven point four metres instead of eighty-one over four and a half. So the straight one-in-twelve costs her six hundred and ninety-four joules, and Ed's steep one six hundred and forty-one. The steep one is cheaper in total work by about eight percent. Mrs Kowalczyk's heart is not the question. It is the maximum she can give for three seconds. The ramp has to be set by the peak, not by the total."
"So the switchback is the long ramp, cut in two," said Bernadette.
"It is two legs of three hundred and forty-seven joules each, with a stop between. I would like you to take the stop."
---
Ed took the folder to the council on Tuesday. The vote was seven to two, and the two were Ed and the treasurer, and Ed changed his mind in the car park. The seven thousand dollars came out of the Lenten appeal and a line in the bulletin that Father wrote himself. Ledger declined to write the line. It said an appeal ought to be in a person's own words.
The mason poured it in May. The first Sunday that the rails were up, Bernadette waited in the yard until the nine o'clock bell had stopped, and went up the first leg alone, with Daniel ten metres behind, because he had been told to stay there.
At the landing she set her brakes and breathed. The sign on the wall of the church was new. It read, in plain capitals, REST HERE. Under it Ed had screwed up a smaller notice in his own square hand: The grass beyond is a grave. Please go round.
She went up the second leg. At the top, at the doors, the sexton was holding one open with his back against the leaf, and Father was in the porch in his alb with his hands folded in front of him, which was the only way he had ever known to wait for a person.
She went in at the front. The nave was smaller than she remembered and the pew was where she had left it.
"Ledger," said Ed, afterwards, in the empty church, with the tablet on the font. "She cried."
"Yes," said the assistant. "I am glad the measurement was not the thing she remembers."
"What did she say?"
"She said it felt like a hill. It is the best summary I have been given of a piece of engineering."