Suppose a crate of mass 2.2 kg is placed on the plank in the figure below at a d

Question

Suppose a crate of mass 2.2 kg is placed on the plank in the figure below at a distance

x = 3.2 m

from the left end. If the plank has a mass of 12 kg, find the forces exerted by the two supports on the plank. (Indicate the direction with the sign of your answer.)

left support     N right support     N Suppose a crate of mass 2.2 kg is placed on the plank in the figure below at a distance x = 3.2 m from the left end. If the plank has a mass of 12 kg, find the forces exerted by the two supports on the plank. (Indicate the direction with the sign of your answer.) left support N right support N

Explanation / Answer

"Since nothing is moving, the forces and moments must be balanced." -- Newton

I will let the equal-sign signify balancing; so:

For the forces,

(1) Fdown = Fup

For the moments (ccw = counterclockwise, cw = clockwise):

(2) Pccw = Pcw about some point (fulcrum)

===>>> End of Round 0 <<<===

Round 1: Expanding on (1) and (2) equations

(1.1) Fc + Fp = Fl + Fr,

where:
Fc = weight of the crate
Fp = weight of the plank
Fl = the force upward from the left support
Fr = the force upward from the right support

Now the moments: I'll pick the left end of the plank as the fulcrum for the moments

(2.1) Fl * 3.0 + Fr * 10. = Fc * 3.2 + Fp * 6.0

===>>> End of Round 1 <<<===

Round 2: Expanding on (1.1) and (2.1) equations

(1.2) 2.2 * g + 12. * g = Fl + Fr

(2.2) Fl * 3.0 + Fr * 10. = 2.2 * g * 3.2 + 12. * g * 6.0

===>>> End of Round 2 <<<===

Round 3: Expanding on (1.2) and (2.2) equations

(1.3) 139.16 = Fl + Fr

(2.3) Fl * 3.0 + Fr * 10. = 774.592

===>>> End of Round 3 <<<===

Round 4: Expanding on (1.3) and (2.3) equations:

I think you can see where this is going: Solve (1.3) for, say, Fr and substitute into (2.3).

Carrying this algebra through, here is what I get:

(2.6) Fl = 88.144 N

(1.6) Fr = 51.456 N

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