Question

In a stamping machine the die has a mass of 30 kg and falls freely a distance of 2 m to strike a metal block. The depth of indentation is 10mm. Assuming that the die does not rebound Determine the average resistance of the metal. You are required to compare and contrast by solving this problem in two ways:
a) by making use of the principle of conservation of momentum and D’Alembert’s principle
b) by making use of the principle of conservation of energy.

Answer 1

Answer to Question #71543, Physics / Mechanics | Relativity

In a stamping machine the die has a mass of $30\,kg$ and falls freely a distance of $2\,m$ to strike a metal block. The depth of indentation is $10\,mm$. Assuming that the die does not rebound determine the average resistance of the metal. You are required to compare and contrast by solving this problem in two ways:

a) by making use of the principle of conservation of momentum and D'Alembert's principle

b) by making use of the principle of conservation of energy.

Solution.

We have:

$$m = 30\,kg - \text{mass of the die}$$ $$h = 2\,m - \text{distance of die falling}$$ $$s = 0.01\,m - \text{depth of indentation}$$

a) Find speed of the die at the moment of touching with block:

$$h = \frac{g t^2}{2}$$ $$t = \sqrt{\frac{2h}{g}}$$ $$v = g t = \sqrt{2gh} = \sqrt{2 \cdot 9.8 \cdot 2} = 6.26\,m/s$$

By D'Alembert's principle:

$$ma = mg - F_{res}$$

where $F_{res}$ is the average resistance of the metal.

Then:

$$a = \frac{dv}{dt}; \quad v = \frac{ds}{dt} \Rightarrow a = \frac{dv}{ds} v$$ $$mvdv = (mg - F_{res}) ds$$ $$\frac{mv^2}{2} = (mg - F_{res}) s$$

Answer to Question #71543, Physics / Mechanics | Relativity

Answer:

$$F_{res} = \left| mg - \frac{mv^2}{2s} \right| = \left| 30 \cdot 9.8 - \frac{30 \cdot 6.26^2}{2 \cdot 0.01} \right| = 58487\,N$$

b) By the principle of conservation of energy:

$$mg(h + s) = F_{res}s$$

Answer:

$$F_{res} = \frac{mg(h + s)}{s} = \frac{30 \cdot 9.8 \cdot (2 + 0.01)}{0.01} = 59094\,N$$

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