Stability: Leaning Horse against a Wall

[Jazz]

Homework Statement

Data:

$m = 500\ kg$
Distances given in the image.

Homework Equations

$\tau = rF\sin(\theta)$

$F_{net} = ma$

The Attempt at a Solution

It seems this problem is intended to be one where torque applies, but I don't see it in that way. And of course my answer doesn't agree with that given by the textbook :).

The diagram:

According to my understanding, there is a component of the weight acting along the horse's body (the diagonal I've drawn) and another one perpendicular to its body, which is making it rotate to the left. I labeled the latter $w_{\perp(horse)}$. The component of $w_{\perp(horse)}$ that lies perpendicular to the wall, I think, has the same magnitude that the force exerted by the wall on the horse. This is what I mean:

Then:

$w_{\perp(horse)} = w\sin(\theta)$

$F_{wall} = w_{\perp(horse)}\cos(\theta)$

$F_{wall} = w\sin(\theta)\cos(\theta)$

And by Newton's Third Law, this is the force exerted on the wall.

The angle is found to be $14.04º$, so

$F_{wall} = 500\ kg \cdot 9.80\ m/s^2 \cdot \sin(14.04º) \cdot \cos(14.04º) = 1.15 \times 10^3 N$

Maybe it should have a negative sign since it's in the opposite direction, but that is not what I'm worried about. The textbook's answer is $1.43 \times 10^3 N$. I have no idea what I'm missing /:.

Thanks !

 

[PhanthomJay]

You are over complicating this problem and getting messed up. Sum torques about the horses feet. Torque of a force is force times perpendicular distance.

 

[Jazz]

You are over complicating this problem and getting messed up. Sum torques about the horses feet. Torque of a force is force times perpendicular distance.

Ah, I see. I confused the perpendicular distance with the perpendicular component of the force (I've been doing so all the time :) ).

$\sum \tau = 0 = \tau_{ccw} - \tau_{cw}$

$\sum \tau = 0 = \tau_{weight} - \tau_{wall}$

$\tau_{wall} = \tau_{weight}$

$F_{wall} = \frac{500\ kg \cdot 9.80\ m/s^2 \cdot 0.35\ m}{1.20\ m}$

$F_{wall} = 1.43 \times 10^3\ N$

If I were to solve a similar problem where $\sum F = 0$, I must then assume $F_{wall} = friction$ and $w = F_{normal}$ about the horses feet, right? I want to be sure about it, because I also got confused with that and with the fact that the forces are acting along different heights/distances.

 

[PhanthomJay]

If I were to solve a similar problem where $\sum F = 0$, I must then assume $F_{wall} = friction$ and $w = F_{normal}$ about the horses feet, right? I want to be sure about it, because I also got confused with that and with the fact that the forces are acting along different heights/distances.

Yes, that is correct. Normal force from ground acts up on the horses feet equal in magnitude to its weight, and the ground friction force on its feet acts left , equal in magnitude to the normal wall force.

 

[Jazz]

Yes, that is correct. Normal force from ground acts up on the horses feet equal in magnitude to its weight, and the ground friction force on its feet acts left , equal in magnitude to the normal wall force.

Nice!

Thanks for helping me [: