Physics, 08.04.2020 01:45, fansofboys

Three heavy rods are all made of the same uniform material. These rods have lengths 3 m, 4 m, and 5 m, and compose the sides of a 3-4-5 right triangle. Find the coordinates of the center of mass of the triangle. (Assume that the right angle is at the origin, and that the two perpendicular sides are parallel to the axes, with the longer leg in the vertical direction.)

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answered: shann7074

[ 2.67 , 1 ] m

Explanation:

Given:-

- The side lengths of the rods are as follows:

a = 4 m , b = 4 m , c = 5 m

a = Base , b = Perpendicular , c = Hypotenuse

- All rods are made of same material with uniform density. With

Find:-

Find the coordinates of the center of mass of the triangle.

Solution:-

- The center of mass of any triangle is at the intersection of its medians.

- So let’s say we have a triangle with vertices at points (0,0) , (a,0) , and (0,b).

Median from (0,0) to midpoint (a/2,b/2) of opposite side has equation:

bx−ay=0

Median from (a,0) to midpoint (0,b/2) of opposite side has equation:

bx+2ay=ab

Median from (0,b) to midpoint (a/2,0) of opposite side has equation:

2bx+ay=ab

Solve all three equations simultaneously:

bx−ay=0 , bx = ay

ay + 2ay = ab , 3ay = ab , y = b/3

bx = b/3

x = a / 3

So the distance from the median to each leg of the triangle is 1/3 length of other leg.

- So the coordinates of the centroid for right angle triangle would be:

[ 2a/3 , b/3 ]

[ 2.67 , 1 ] m

answered: Guest

it is a negative acceleration

answered: Guest

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

$x(t)=ax(x-6)$

where $a$ is unknown. because the position is parabolic, there is symmetry about the midpoint at $t=3$ , and we know that

$x(3)=10\implies -9a=10\implies a=-\dfrac{10}9$

so that the position function is

$x(t)=-\dfrac{10}9x(x-6)$

which means the velocity is

$v(t)=\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac{20}9x+\dfrac{60}9$

then the total distance traveled is

$\displaystyle\int_{t=0}^{t=6}|v(t)|\,\mathrm dt=\left\{\int_{t-0}^{t=3}-\int_{t=3}^{t=6}\right\}v(t)\,\mathrm dt=20$

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at $x=0$ and ends at $x=0$ , so its displacement is 0.

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Three heavy rods are all made of the same uniform material. These rods have lengths 3 m, 4 m, and 5...