A steam hammer falls from a height of h = 3 m onto a brass blank. How many times n must it fall for the temperature

A steam hammer falls from a height of h = 3 m onto a brass blank. How many times n must it fall for the temperature of the ingot to rise by ∆T = 19.87 K. Heating the ingot consumes 60% of the heat released during impacts. The specific heat capacity of brass is c = 400 J / kg ⋅ K. The mass of the hammer is M = 5 tons, the mass of the ingot is m = 200 kg.

h = 3 m.

g = 9.8 m / s2.

C = 400 J / kg * ° K.

M = 5 t = 5000 kg.

m = 200 kg.

∆T = 19.87 K.

Efficiency = 60%.

n -?

Let us find the amount of heat energy Qb required to heat the brass bar Qb = C * m * ∆T, where C is the specific heat capacity, m is the mass of the bar, ∆T is the temperature change of the bar.

With one fall of the hammer, efficiency = 60%, its potential energy is converted into the internal energy of a brass bar: M * g * h * efficiency / 100% = Q1.

Qb = n * Q1.

C * m * ∆T = n * M * g * h * efficiency / 100%.

We find the number of hammer blows by the formula: n = C * m * ∆T * 100% / M * g * h * efficiency.

n = 400 J / kg * ° K * 200 kg * 19.87 K * 100% / 5000 kg * 9.8 m / s2 * 3 m * 60% = 18.

Answer: to heat a brass bar, you need to make n = 18 blows with a hammer.