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kq6up
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Homework Statement
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A block of mass m slides down an incline with a height h. Later, it collides with a spring and compresses the spring to some maximum displacement. How long does it take to reach maximal compression.
Homework Equations
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1) ##mgh=\frac{1}{2}mv^2+\frac{1}{2}kx^2##
2) ##
t=\int _{0}^{ x_{max}}{\frac{1}{v}}{dx }
##
The Attempt at a Solution
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By conservation of energy:
##mgh=\frac{1}{2}mv^2+\frac{1}{2}kx^2## Rearranging to solve for ##v##:
##v=\sqrt{2gh-\frac{k}{m}x^2}##
Inserting into 2):
##t=\int _{0}^{ x_{max}}{\frac{1}{\sqrt{2gh-\frac{k}{m}x^2}}}{dx }##
This is in the form:
##\int { \frac { 1 }{ \sqrt { a^{ 2 }-u^{ 2 } } } } { du }=\sin ^{ -1 } \frac { u }{ a } ##
Using the appropriate substitution we get:
##t=\sqrt { \frac { m }{ k } } \left[ \sin^{-1} \frac{u}{a} \right]_{0}^{\sqrt{\frac{k}{m}}x} ##
Where: ##a=\sqrt{2gh}## and ##x_{max}=\sqrt{\frac{2mgh}{k}}##
Yields the result:
##t=\sqrt{\frac{m}{k}}\frac{\pi}{4}##
Does this look correct? The result is independent of gravity and height of the initial mass.
Thanks,
kQ6Up
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