An 80 kg person stands on a uniform 5.0 kg ladder that is 4.0 m long, as shown.

Question

An 80 kg person stands on a uniform 5.0 kg ladder that is 4.0 m long, as shown. The floor is rough; hence it exerts both a normal force, f1, and a frictional force, f2, on the ladder. The wall, on the other hand, is frictionless; it exerts only a normal force, f3. Using the dimensions in the figure, find the magnitudes of f1, f2, and f3. f1 = N f2 = N f3 = N

An 80 kg person stands on a uniform 5.0 kg ladder that is 4.0 m long, as shown. The floor is rough; hence it exerts both a normal force, f1, and a frictional force, f2, on the ladder. The wall, on the other hand, is frictionless; it exerts only a normal force, f3. Using the dimensions in the figure, find the magnitudes of f1, f2, and f3. f1 = N f2 = N f3 = N

Explanation / Answer

These sorts of static problems can generally be solved by using two simple principles:

(1) Total force on the system is 0.

(2) Total torque on the system is 0.

The system in this case is the (massless) ladder.

Of course we must treat the forces as vectors in which case the mathematical vector expression of (1) and (2) is:

(1) ? F = 0

(2) ? ? = ? r X F = 0

If we take horizontal as i and vertical as j then:

(f2 - f3) i + (f1 - mg) j = 0 (1)

Taking the torque about the base of the ladder:

r1 X (-mg) j + r2 X (-f3) i = 0 (2)

using the determinant form of the cross product:

|i ......j ......k| = r1 X (-mg) j = -(r1x)(mg) k

|r1x.. r1y ...0|

|0... -mg.... 0|

|i ......j .....k| = r2 X (-f3) i = (r2y)(f3) k

|r2x.. r2y ..0|

|-f3.... 0.... 0|

so that:

( (r2y)(f3) - (r1x)(mg) ) k = 0 (2)

summary:

(f2 - f3) i + (f1 - mg) j = 0 (1)

( (r2y)(f3) - (r1x)(mg) ) k = 0 (2)

Here we have 3 equations:

f2 - f3 = 0 (3)

f1 - mg = 0 (4)

(r2y)(f3) - (r1x)(mg) = 0 (5)

of course, from the diagram r1x = 0.7 and r2y = 3.8 and mg = 80*9.81 = 784.8 so the last two equations are:

f1 - 784.8 = 0 (4*)

3.8f3 - .7* 784.8 = 0    ====   3.8f3- 549.36=0 (5*)

f2-f3=0 (3)

equations (3), (4*) and (5*) are now 3 simultaneous equations for 3 variables f1, f2 and f3. Solve to give:

f1 = 784.8 N

f2 = 144.57 N

f3 = 144.57 N

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