2 masses and 1 spring jumping object question

[alldan25889]

Here is a question as part of my coursework.
A spring of stiffness k connects a part of a jumping robot that rests on the ground of mass M1 to a free mass M2. When fully compressed the spring has length Lc and relaxed has length Lr. Sping has negligible mass and can be streched and compressed...
Questions:
a) Assuming 100% efficiency, to what height will the robot's centroid jump above where it started?
b) Derive two equations that describe vertical acceleratiosn of the two masses and derive a third that relates their instantaneous heights by the length of the spring
c) If solving these equations, what additional conraints would have to be applied to the instantaneous height of M1?

I have had a go at the first question and equated the (m1+m2)gh = 1/2 k (Lr-Lc)^2. I know that the value for h is not exactly what would be the height. For question 2 I attempted isolated free body diagrams for both masses and obtained the following expressions. m2 x a2 = k x (extension) and m1 = -m1g + k (extension). Where extension = height traveled by m2 - height traveled by m1.

Would appreciate your opinions.
Allen

 

[jbsteele]

For question a), the answer to your equation is the maximum height that the centroid of the robot will reach, assuming perfect efficiency. For question b), you are correct that you need two equations to describe the vertical accelerations of the two masses, and a third equation that relates their instantaneous heights by the length of the spring. The two equations for the accelerations should be:m1*a1 = -m1*g + k*(Lr-Lc) m2*a2 = k*(Lr-Lc) The third equation relating the instantaneous heights is derived from the conservation of energy, which states:m1*gh1 + m2*gh2 = 1/2*k*(Lr-Lc)^2 This equation can be rearranged to solve for the difference in heights of the two masses as a function of the length of the spring:h2 - h1 = (Lr-Lc)^2 / 2g For question c), the additional constraint that would need to be applied to the instantaneous height of M1 is that it must remain at or above the initial height. This means that the value of h1 should never be negative, since it is the difference in heights of the two masses that is being calculated.