A block of mass m is placed on a horizontal frictionless surfaceand between two

Question

A block of mass m is placed on a horizontal frictionless surfaceand between two vertical walls. The mass is connected to the wallon the left by a spring with the spring constant constantk1. It is also connected to the wall on the right by aspring constant k2. The system is in equilibrium whenplaced half-way between the walls. When the mass is pushed awayfrom this point, it oscillates. a) What is the frequency ofoscillations? b) What would be the frequency of oscillations whenthe system is placed on the Moon? c) The mass is moved by thedistance A to the right and released at rest. What is themaximum velocity of the mass? d) Continuing part c): at whatdistance away from the center is the total energy of the systemevenly distributed between its kinetic and potentialcontributions?

Explanation / Answer

a) What is the frequency of oscillations? when the block is displaced by x the restoring force is(k1+k2)x hence the frequency of oscillation is (k1+k2)/m b) What would be the frequency of oscillations when thesystem is placed on the Moon? it is not at all related to gravity at that point and does notdepend on it hence the same value c) The mass is moved by the distance A to the rightand released at rest. What is the maximum velocity of the mass? this is A as A is the amplitude of the oscillation weknow the angular frequency d) Continuing part c): at what distance away from the centeris the total energy of the system evenly distributed between itskinetic and potential contributions? this is when the velocity is 1/2 times that of maximumhence A/2=A*A-x*x now solve for x

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