A crate of mass m falls from a height h onto the end of a platform, as shown in the figure. The spring is initially unstretched and the mass of the platform can be neglected. Assuming that there is no loss of energy, the maximum elongation of the spring is:
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Given,
Mass = m
height = h
let the elongation of spring = x
According to the law of conservation of energy,
potential energy = spring energy stored
=> m(h+x) = kx²/2
=> kx² - 2mx - 2mh = 0
which is a quadratic equation, solving for x we get,
x = [2m ±√(4m² -4.k.-2m)]/2k
x = [2m ±√(4m² + 8km)]/2k
cancelling 2 from both the numerator and denominator we get,
=> x = [m ±√(m² +2km)]/2k
which is the required expression for the maximum elongation of the spring.
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