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This one is killing me.
1. Homework Statement
A SHO is resting on a horizontal surface with which it has static and
kinetic friction coefficients us and uk . We have k for the spring constant
and m for the mass. The surface is actually part of a conveyor belt
that is maintained at a speed u whatever the behavior of the SHO. (speed is to the right)
see figure
Assume that the mass of the SHO is initially at rest at its equilibrium
position. The motion can be divided into distinct regimes, depending
upon whether there is slipping or not. You are to begin with the first
regime.
(a) Derive expressions for the displacement and velocity as a function
of time.
(b) Derive the condition under which the SHO will oscillate sinu
soidally forever.
Consider the situation under which the mass can experience static
friction.
(c) At what time t1 does the SHO lose its sinusoidal behavior and
sticks? Express t1 in terms of the given parameters.
(d) What is the displacement at the point when the mass sticks? Ex
press your answer in terms of u, ?k , g, and the angular frequency
?0 of the SHO.
(e) What is the motion of mass when it sticks?
(f ) What is the displacement when the mass begins to slip?
2. The attempt at a solution
I'm really only having trouble with the first part. I'm thinking that the conveyor belt picks up the mass due to static friction and carries it with constant speed until kx overcomes static friction at which point there is slipping, etc. Is this correct? and if so how do we explain how it is initially put into motion because we cannot have an instantaneous force brining the mass to u.
Thanks
1. Homework Statement
A SHO is resting on a horizontal surface with which it has static and
kinetic friction coefficients us and uk . We have k for the spring constant
and m for the mass. The surface is actually part of a conveyor belt
that is maintained at a speed u whatever the behavior of the SHO. (speed is to the right)
see figure
Assume that the mass of the SHO is initially at rest at its equilibrium
position. The motion can be divided into distinct regimes, depending
upon whether there is slipping or not. You are to begin with the first
regime.
(a) Derive expressions for the displacement and velocity as a function
of time.
(b) Derive the condition under which the SHO will oscillate sinu
soidally forever.
Consider the situation under which the mass can experience static
friction.
(c) At what time t1 does the SHO lose its sinusoidal behavior and
sticks? Express t1 in terms of the given parameters.
(d) What is the displacement at the point when the mass sticks? Ex
press your answer in terms of u, ?k , g, and the angular frequency
?0 of the SHO.
(e) What is the motion of mass when it sticks?
(f ) What is the displacement when the mass begins to slip?
2. The attempt at a solution
I'm really only having trouble with the first part. I'm thinking that the conveyor belt picks up the mass due to static friction and carries it with constant speed until kx overcomes static friction at which point there is slipping, etc. Is this correct? and if so how do we explain how it is initially put into motion because we cannot have an instantaneous force brining the mass to u.
Thanks
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