uniform plank walk

**rcmango** · November 30th 2007, 10:15 PM

A uniform plank of length 4.9 m and weight 210 N rests horizontally on two supports, with 1.1 m of the plank hanging over the right support (see the drawing). To what distance x can a person who weighs 453 N walk on the overhanging part of the plank before it just begins to tip?
in meters

**kalagota** · December 1st 2007, 05:39 AM

Originally Posted by rcmango

A uniform plank of length 4.9 m and weight 210 N rests horizontally on two supports, with 1.1 m of the plank hanging over the right support (see the drawing). To what distance x can a person who weighs 453 N walk on the overhanging part of the plank before it just begins to tip?
in meters

(since no one answers it yet and it is placed in the Urgent HW, i'll try it..)
i can't imagine the drawing.. Ü maybe something like this..

Code:

 o                      o
 o                      o
 o                      o
''''''''''''''''''''''''''''''''''''''''''

do you have the answer.. actually, i got 0.6258 m from the right support.. well, not sure on this that is why i don't put the solution.. can i see what you're doing?

**rcmango** · December 3rd 2007, 10:22 AM

Your answer is correct, I could not figure this one out.

looks alot like this, plank - View Image - Post Your Image

can you please explain how i am supposed to set this up.

**kalagota** · December 5th 2007, 04:48 AM

Originally Posted by rcmango

Your answer is correct, I could not figure this one out.

looks alot like this, plank - View Image - Post Your Image

can you please explain how i am supposed to set this up.

good.. now, since the plank has uniform weight $w_p$ , let's assume that it is concentrated at the middle of the plank..

so, if the length of the plank is L, then the weight is located at $\frac{L}{2}$ ..

the distance of the left support from the middle of the plank is $\frac{L}{2} - 1.1m$ ..

now, if $w_{man}$ is the weight of the man, then the product of $w_p$ and $\frac{L}{2} - 1.1m$ (the distance of the middle of the plank from the right support) must be equal to the product of $w_{man}$ and the distance, $x$ , of the man from the right support..

that is, $w_p \cdot \left( \frac{L}{2} - 1.1 \right) = x \cdot w_{man}$ ..

now, substitute the values.. Ü

**topsquark** · December 5th 2007, 05:52 AM

Originally Posted by rcmango

Your answer is correct, I could not figure this one out.

looks alot like this, plank - View Image - Post Your Image

can you please explain how i am supposed to set this up.

This is just going to mimic kalagota's response, but let me put it in "Physics" terms for you.

The plank is not rotating, so it is in static equilibrium. This denotes not only a = 0 for the system, but also means the system doesn't rotate.

So let's pick a coordinate system where +y is upward. According to the diagram we have a force $S_L$ by the left support acting straight upward, a force $S_R$ by the support on the right acting straight upward. Both of the support forces act on the board at the "point" of the support. We also have the man's weight $w_m$ acting straight down from where the man is standing (x meters to the right of the right support), and a weight $w_p$ of the plank acting straight down from the geometric center of the plank.

Newton's 2nd tells us
$\sum F_y = S_L + S_R - w_m - w_p = 0$

or
$S_L + S_R - 453 - 210 = 0$

Now, we have two unknown forces and one unknown x. So we need two more equations.

Let's pick an axis of rotation at the point where the support on the right meets the plank and call this point A. The system is not rotating about point A, so
$\sum \tau _A = 0$

$\sum \tau _A = -S_L(4.9 - 1.1) - w_p(2.45 - 1.1) + w_mx = 0$
(Recall that a counterclockwise torque is negative by default.)

$-3.8S_L - 283.5 + 453x = 0$

We still need one more equation. The nice thing about static equilibrium is that the system doesn't rotate about any point, so we are free to pick whatever axis of rotation we desire, whether it is a physical axis of rotation or not. So I'm going to pick another axis of rotation B located where the support on the left meets the plank. Thus
$\sum \tau _B = w_p(2.45) - S_R(4.9 - 1.1) + w_m(4.9 - 1.1 + x) = 0$

or
$514.5 - 3.8S_R + 453(3.8 + x) = 0$

So we now have three equations in three unknowns:
$S_L + S_R - 453 - 210 = 0$
and
$-3.8S_L - 283.5 + 453x = 0$
and
$514.5 - 3.8S_R + 453(3.8 + x) = 0$

Solving the system I get $x = 0.625828~m$

-Dan

**kalagota** · December 5th 2007, 05:57 AM

you're really a physics maestro..
i was about to that solution before but when i realized that there was in need to solve a system of equation, i became passive, so i tried the simplest way to solve it..

(whoa, it made me remember topics of the first of the two physics course i had..) Ü

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