Question

A raccoon has climbed a building downtown where the raccoon finds a beam that has been installed to hold a sign for a business. The uniform beam, which is 3.2 meters long and has a weight of 225 N, is attached to the wall of the building by a hinge and is supported by a light wire that makes an angle of 30.0° above the horizontal. The raccoon, who has a mass of 25 kg, walks onto the beam, not knowing that the greatest tension that the wire can support without breaking is 650 N. To what maximum distance (x) can the raccoon walk on the beam without breaking the wire? When he reaches that point, what are the horizontal and vertical components of the force exerted on the beam by the hinge?

30° – 3.2 m

Answer 1

Tsin30 Rx 30 deg 1.6m Tcos30 Wb

Given:

Weight of the beam, $W_{b}=225N$

Length of the beam, $l=3.2m$

Mass of the raccoon,

m, = 25kg

Maximum tension supported by the wire,

T = 650N

are the vertical and horizontal components of the force exerted on the beam by the hinge.

=========

Find

-----------

Take moment about the hinge

> M =0

$Tsin30*l-W_{b}*\frac{l}{2}-W_{r}*x=0$

$Tsin30*l-W_{b}*\frac{l}{2}-m_{r}*g*x=0$

$650N*sin30*3.2m-225N*\frac{3.2m}{2}-25kg*9.81m/s^{2}*x=0$

$650*sin30*3.2-225*\frac{3.2}{2}-25*9.81*x=0$

$1040-360-25*9.81*x=0$

$1040-360=25*9.81*x$

$680=25*9.81*x$

680 25 * 9.81 = C

$x=2.77268m$

ANSWER: ${\color{Red} x=2.8m}$

===========================

The horizontal and vertical components of the force exerted on the beam by the hinge

Consider all vertical forces

$\sum F_{y}=0$

$R_{y}+Tsin30-W_{r}-W_{b}=0$

$R_{y}+Tsin30-m_{r}*g-W_{b}=0$

$R_{y}=m_{r}*g+W_{b}-Tsin30$

$R_{y}=25kg*9.81m/s^{2}+225N-650N*sin30$

$R_{y}=25*9.81+225-650*sin30$

ANSWER: Vertical component of force at hinge, ${\color{Red} R_{y}=145.25N}$

=================

Consider all horizontal components

$\sum F_{x}=0$

$R_{x}-Tcos30=0$

$R_{x}=Tcos30$

$R_{x}=650N*cos30$

ANSWER: Horizontal component of force at hinge, ${\color{Red} R_{x}=562.92N}$

==============================================