Question

A 61 g ice cube can slide without friction up and down a 31 ∘ slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring 12 cm . The spring constant is 20 N/m . When the ice cube is released, what distance will it travel up the slope before reversing direction?

Answer 1

for a simple approach, find the equilibrium extension, ie the extension as if were the mass at rest on the spring. oscillation should be symmetrical about that point.


more detailed approach:


energy of system to start with:
\(E_o = \dfrac{1}{2}kx^2 = \dfrac{1}{2}(20)(0.12)^2\)

when the cube has moved a distance x up the slope

\(E(x) = mg x \sin 31 + \dfrac{1}{2}k(0.12-x)^2 + \dfrac{1}{2}m \dot x^2 = E_o\)
\(\dot E(x) = mg \color{red}{\dot x} \sin 31 + 2\dfrac{1}{2}k(0.12-x)(-\color{red}{\dot x}) + 2\dfrac{1}{2}m \color{red}{\dot x} \ddot x = 0\)
\(\implies mg \sin 31 - k(0.12-x)+m\ddot x = 0\)

tidy up and solve the DE.