Map Sapling Learning A small mouse has gotten into a peculiar situation. Somehow

Question

Map Sapling Learning A small mouse has gotten into a peculiar situation. Somehow, the mouse has found himself hanging off the end of the metal structure shown in the figure below. Hopefully the structure can support the 26.6 g mouse because it would not make the fall. The structure is rigidly fixed to a wall and has mass per unit length of 2.91 g/cm and dimensions x 18.6 cm and y-34.6 cm, defined as y = 0.1x2 (where the attachment point P has coordinates Xp = 0, yP = 0). Calculate the torque about the attachment point P. Number =110.0978 | N. m ncorrect

Explanation / Answer

Equation of Curve is Y = 0.1 x^2

Torque net = Torque Due to mouse + Torque Due to metal

Tmouse = F r = m g r

Tmouse = 0.0266 * 9.81 * 0.186   = 0.0485 N m

Torque Due to metal

let dl be the small length of the metal which subtends length dx on x axis

mass of element dl = dm = L dl

dT = d mg x

Tmetal = integration of dt = int from 0 to 18.6 cm of L dl g x

Tmetal = L g int from 0 to 18.6 cm of x dl

also dl^2 = dx^2 + dy^2

dl^2 = dx^2 *(1 + (dy/dx)^2)

dl^2 = (1 + dy/dx) dx

from y = 0.1 x^2

dy = 0.2 x dx

dy/dx = 0.2 x

dl^2 = 1 + (0.2x)^2 dx

dl = (1 + 0.04 x^2) dx

Tmetal = L g * intgetratuon of x*sqrt(1 + 0.04x^2) dx

let z = 1+ 0.04 x^2

dz = 0.08 xdx

xdx = dz/0.08

from limits

x =0    --> z =1

x = 18.6 cm z = 14.83 cm

Tmetal = L g * int from1 to 14.83 cm   sqrt z dz/0.08

Tmetal = Lg/0.08 * (z^3/2)/3/2 from 1 to 14.83 cm

Tmetal = L g /0.08 * 2/3 * (14.83^(3/2) - 1^)3/2)

Tmetal =    2.91*9.81/0.08 * 2/3 * (57.15 - 1)

Tmetal = 0.133 Nm

Tnet = T mouse + Tmetal

Tnet = 0.0485 + 0.1335   = 0.182 Nm
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