Question

A 3000-kg crane is supporting a 10,000-kg crate. The crane pivots about point A and is at rest pressed against a support at B. (a) Find the force acting on the crane at point A. (b) Find the force acting on the crane at point B. (c) Use Order of Magnitude sense-making to determine the reasonableness of your answers to parts (a) and (b). Hint: consider how t

Answer 1

The answers are physically and mathematically reasonable, since both have the same order of magnitude of the external forces shown in the figure.

Procedure - Determination of forces acting on a rigid body

Let be a system at equilibrium, which mathematically is represented by the following formulas:

$\Sigma F = 0$ (1)

$\Sigma M = 0$ (2)

a) Force acting on the crane at point A

We construct equations around points A and B by Newton's Laws and D'Alembert Principle:

Point A

$\Sigma M = F_{B}\cdot (1\,m)-(3000\,kgf)\cdot (2\,m)-(10000\,kgf)\cdot (6\,m) = 0$ (3)

Point B

$\Sigma M = -F_{A,x}\cdot (1\,m) - (3000\,kgf)\cdot (2\,m)-(10000\,kgf)\cdot (6\,m) = 0$ (4)

Entire system

$\Sigma F_{x} = F_{A,x} + F_{B} = 0$ (5)

$\Sigma F_{y} = F_{A,y} -3000\,kgf-10000\,kgf = 0$ (6)

The solution of the entire system is: $F_{A,x} = -66000\,kgf$, $F_{B} = 66000\,kgf$ and $F_{A,y} = 13000\,kgf$.

The magnitude of the force acting on the crane at point A is determined by the Pythagorean theorem:

$F_{A} = \sqrt{(-66000\,kgf)^{2}+(13000\,kgf)^{2}}$

$F_{A} \approx 67268.120\,kgf$

The force acting on the crane at point A has a magnitude of approximately 67268.128 kilograms-force. $\blacksquare$

b) The force acting on the crane at point B

The force acting on the crane at point B has a magnitude of approximately 66000 kilograms-force. $\blacksquare$

c) Order of Magnitude sense-making

The answers of parts (a) and (b) have an order of magnitude of $10^{3}$, the same order of magnitude of the external forces shown in the figure. Hence, those answers are physically and mathematically reasonable.

Remark

Figure is missing. The statement is incomplete. Complete statement is presented below:

A 3000 kilograms-force is supporting a 10000 kilograms-force crate. The crane pivots about point A and is at rest pressed against a support at B.

(a) FInd the force acting on the crane at point A.

(b) Find the force acting on the crane at point B.

(c) Use order of magnitude sense-making to determine the reasonableness of your answers to parts (a) and (b). Hint: consider how the lever arm to the crate  is much different than that to other points.

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