On a smooth surface, a mass `m_1=2 kg` is attached to the first object by a spring with spring constant `k_1=4 N/m` . Another mass `m_2=1 kg` is attached to the first object by a spring with spring...
On a smooth surface, a mass `m_1=2 kg` is attached to the first object by a spring with spring constant `k_1=4 N/m` . Another mass `m_2=1 kg` is attached to the first object by a spring with spring constant `k_2=2 N/m` . The objects are aligned horizontally so that the springs are their natural lengths. If both objects are displaced `3 m` to the right of their equilibrium positions and then released, what are the equations of motion for the two objects?
The force on block 1 from the first spring is
Where `x` is the position to the right of the equilibrium position. Let y be the distance to the right of block 2's equilibrium position. Then the second spring exerts a force `F_2` on block 1 given by
Think about it. If `x=y` there should be no force from spring 2. Also if `ygtx` then there should be a pull to the right (positive) on block 1. If `xlty` then the block should pull to the left (negative).
Since block 2 is only attached to spring 2, from Newton's third law
Now apply Newton's second law to both the blocks.
Plug in the values for `k` and `m` , then move the terms to the left hand side.
I'm going to set let `D:=d/dt` and solve this system of equations by the elimination method.
Now multiply eq. `(1)` by `(D^2+2)` and multiply eq. `(2)` by `2` .
Add eq. `(1)` and eq. `(2)` together to eliminate `y`.`(D^2+2)(2D^2+6)x-2(D^2+2)y+2(D^2+2)y-4x=0 `
Try a solution of the form `x(t)=e^(rt)` .
Then eq. `(3)` takes the form
Solve for the roots of this characteristic equation.
The roots are `r=i, -i, 2i, -2i` .
Using Euler's formula, it follows that two linearly independent solutions are
The general solution is a superposition of the imaginary and real parts of the linearly independent solutions.
To find `y(t)` use earlier equation to put `y` in terms of `x` ,
Differentiate the general solution for `x(t)` twice and plug it in. You will find
To determine the coefficients apply the initial conditions:
This will yield a system of 4 equations. You should find that,
x(t) is in black and y(t) in red.