A $51\,\text{kg}$ person stands at a distance $d=1.5\,\text{m}$ from the end of a $L=3.9\,\text{m}$ long, $10\,\text{kg}$ scaffolding plank The plank is balanced on two pedestals at points A and B. Compute the magnitude of the normal forces on the plank at points A and B. Assume $g=9.8\,\text{m/s}^2$ to answer this question and present your answers to two significant figures.

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Grade 6

Find a value using two-variable equations: word problems

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Q. A

51\,\text{kg}

person stands at a distance

d=1.5\,\text{m}

from the end of a

L=3.9\,\text{m}

long,

10\,\text{kg}

scaffolding plank The plank is balanced on two pedestals at points A and B. Compute the magnitude of the normal forces on the plank at points A and B. Assume

g=9.8\,\text{m/s}^2

to answer this question and present your answers to two significant figures.

Calculate Person's Weight: The person's weight (force due to gravity) is $F = m \times g$ , where $m$ is the mass and $g$ is the acceleration due to gravity. $\newline$ Calculate the person's weight. $\newline$ $F = 51\,\text{kg} \times 9.8\,\text{m/s}^2 = 499.8\,\text{N}$
Calculate Plank's Weight: The plank's weight is also $F = m \times g$ . Calculate the plank's weight. $F = 10\,\text{kg} \times 9.8\,\text{m/s}^2 = 98\,\text{N}$
Calculate Total Weight: The total weight acting on the plank is the sum of the person's weight and the plank's weight. $\newline$ Total weight = $499.8\,\text{N} + 98\,\text{N} = 597.8\,\text{N}$
Calculate Moment Person's Weight: The plank is in static equilibrium, so the sum of moments about any point is zero. $\newline$ Take moments about point A to find the force at point B. $\newline$ Let $x$ be the distance from A to the person's position. $\newline$ $x = L - d = 3.9\,\text{m} - 1.5\,\text{m} = 2.4\,\text{m}$
Calculate Moment Plank's Weight: Calculate the moment due to the person's weight about point A. $\newline$ Moment = Force * distance = $499.8\,\text{N} * 2.4\,\text{m} = 1199.52\,\text{Nm}$
Calculate Normal Force at B: Calculate the moment due to the plank's weight about point A. $\newline$ The plank's center of gravity is at its midpoint, which is $L/2$ from A. $\newline$ Moment = Force $\times$ distance = $98\,\text{N} \times (3.9\,\text{m}/2) = 98\,\text{N} \times 1.95\,\text{m} = 191.1\,\text{Nm}$
Calculate Normal Force at A: Sum of moments about A is equal to the moment due to the normal force at B ( $N_B$ ) times the distance $L$ .
Sum of moments = $N_B \times L$
$1199.52\,\text{Nm} + 191.1\,\text{Nm} = N_B \times 3.9\,\text{m}$
$N_B = (1199.52\,\text{Nm} + 191.1\,\text{Nm}) / 3.9\,\text{m}$
$N_B = 1390.62\,\text{Nm} / 3.9\,\text{m}$
$N_B = 356.57\,\text{N}$
Calculate Normal Force at A: Sum of moments about A is equal to the moment due to the normal force at B ( $N_B$ ) times the distance $L$ .
Sum of moments = $N_B \times L$
$1199.52\,\text{Nm} + 191.1\,\text{Nm} = N_B \times 3.9\,\text{m}$
$N_B = (1199.52\,\text{Nm} + 191.1\,\text{Nm}) / 3.9\,\text{m}$
$N_B = 1390.62\,\text{Nm} / 3.9\,\text{m}$
$N_B = 356.57\,\text{N}$ The total normal force is the sum of the forces at A and B, which is equal to the total weight.
$N_A + N_B = \text{Total weight}$
$N_A + 356.57\,\text{N} = 597.8\,\text{N}$
$N_A = 597.8\,\text{N} - 356.57\,\text{N}$
$L$ $0$