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For horizontal equilibrium
F4 = F2 = 26133 newtons
For vertical equilibrium
F3 = F1 = M*g = 2000kg*9.8m/s^2, or
F3 = 19600 newtons
The magnitude of the force = sqrt(F4^2 + F3^2), or
Magnitude = sqrt(26133^2 + 19600^2), or
Magnitude = 32666 newtons
Therefore, the magnitude of the force is 32666 newtons.
The angle of the force is
angle = atan(19600 newtons/26133 newtons) in degrees
angle = 36.9 degrees north of east
A ladder is sitting on a horizontal floor and leaning against a smooth wall. A bucket of paint is hanging from a rung 80-percent up from the bottom of theladder. The parameters are as follows:
Draw a vector diagram of the forces acting on the ladder.
Solution:
There are five forces acting on the ladder:
The ladder is in static equilibrium.
The coefficient of static friction between the ladder and the wall is zero.
What is the minimum possible value for the coefficient of static friction between the ladder and the floor.
F1 = ?
F2 = m*g = 2.5kg * 9.8m/s^2 = 24.5 newtons
F3 = M*g = 12kg*9.8m/s^2 = 117.6 newtons
F4 = F5*u
For vertical equilibrium,
F5 = F2 + F3 = (2.5kg + 12kg)*9.8m/s^2 = 142 newtons
For rotational equilibrium, the sum of the torques about the bottom of the ladder must be zero.
Compute the horizontal distance to the line of action of the weight of the ladder.
X1 = (L/2)*cos(50 degrees), or
X1 = ((5/2)m)*cos(50 degrees), or
X1 = 1.61m
Compute the horizontal distance to the line of action of the weight of the bucket of paint.
X2 = 0.8*5m*cos(50 degrees), or
X2 = 2.57m
Compute the vertical distance to the line of action of the horizontal force at the top of the ladder.
Y1 = 5m*sin(50 degrees), or
Y1 = 3.83m
Compute sum of the torques about the bottom of the ladder.
Y1*F1 - X2*F2 - X1*F3 = 0, or
F1 = (X2*F2 + X1*F3)/Y1, or
F1 = (2.57m*24.5 newtons+ 1.61m*117.6 newtons)/3.83m
F1 = 65.87 newtons
For horizontal equilibrium,
F4 = 65.87 newtons
By substitution,
F4 = F5*u = 65.87 newtons
u = (65.87 newtons)/F5, or
u = (65.87 newtons)/142 newtons, or
u = 0.46
While the coefficient of static friction could be higher than this and still achieve static equilibrium, if the coefficient were any lower, the ladder wouldslide away from the wall.
I encourage you to repeat the computations that I have presented in this lesson to confirm that you get the same results. Experiment withthe scenarios, making changes, and observing the results of your changes. Make certain that you can explain why your changes behave as they do.
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