Qusestion Statement: For a system of two masses the distance between two masses
ID: 1816972 • Letter: Q
Question
Qusestion Statement: For a system of two masses the distance between two masses M1 and M2 is (A+x2) Where A is a constant. The Springs are undeflected when x1=x2=0. FInd the differential equations for system. Here is the given solution, But i dont understand where the constant A has gone.
Picture missed the ground but its assumed to be below both masses. You can see that the solution talks about " x2 is the relative displacement of M2 WRT M1". I dont understand the spring foces being k1x1 and not something with the constant in them. Kinda confused with the damping forces B as well. Any Help would be greatly appreciated! Thanks!
Explanation / Answer
I hope that you are familiar with basic differentiations as the solution that you have posted involves a bit of calculus. Here I would use x1^{..} as acceleration of a particle and x1^{.} as velocity of the particle.
Since we are considering mass M1 to be moving in positive x direction and x1 being the variable of motion of M1 so we write forces experienced on mass M1 as
M1*acceleration(which is =x1^{..})= -K1*x1+ K2*X2,
this is same as the equation 1 posted by you.
since no other force besides spring force are acting on M1.
As spring forces are proportional to amount of displacement (or deformation) in springs therefore there is no need of any constant term 'A' in equation.
For the mass M2 there are three forces acting, namely - spring force, F(t) which is given in problem and also a drag force since it is fixed on a ground. Actually drag forces are generally expressed as F = B * (velocity of the particle), where B is an inherent constant for any particular type of surfaces.
Now, we write the equation of motion for the mass M2 as
M2*x2^{..}= f(t)-k2*(relative displacement, which = x2) - B*(velocity of Mass M2 with respect to ground).
Now, since we know that V21(relative velocity of 2 with respect to 1 = x2^{.})= V2(velocity of 2 with respect to ground = unknown) -V1(velocity of 1 with respect ground=x1^{.}), therefore we get V2= x2^{.}+{x1^{.}. Substituting these values in the expression yields you with the required expression.
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