Physics Quiz 8

[Ether]

A massless spring of free length L and spring constant K is stretched until a desired force F is achieved. A mass M is attached to the stretched spring, and the spring is then allowed to accelerate the mass until the spring length is L2.

Find a closed-form solution for the speed of the mass when the spring length is L2 (where L2>L). Ignore friction and gravity. Use the solution to explore the effect of small changes in L and K (with F, M, and L2 held constant).

[lemiant]

Energy in the system = F^2 / 2K
Length when stretched (L3) = L + F/K

Potential energy at L2 = 0.5K*(L2-L)^2

Kinetic energy @L2 = F^2 / 2K - 0.5K*(L2-L)^2

Velocity@L2 = Sqrt(2/M( (F^2 / 2K) - 0.5K*(L2-L)^2 ))
=Sqrt(2K/M( (F/K)^2 - 0.5(L2-L)^2))

I don't know what to do with that.

[dan2915]

I think it should be v=sqrt( F^2/(k*m)-(k/m)*(L2-L)^2).

[Ether]

Originally Posted by dan2915

I think it should be v=sqrt( F^2/(k*m)-(k/m)*(L2-L)^2).

Could you show your work please?

[dan2915]

Sure. I used a somewhat roundabout method. Let A = F/k be the amplitude of the oscillation. The position will follow simple harmonic motion with equation x = L + A * cos(sqrt(k/m)*t), with x = 0 as the left end of the spring and the mass started at t = 0. So we set x = L2 and so L2-L = A * cos(sqrt(k/m)*t). Also, v = x' = - A * sqrt(k/m) * sin(sqrt(k/m)*t). So then we have v^2 = A^2 * (k/m) * sin^2(sqrt(k/m)*t) = (k/m) * (A^2- A^2 * cos^2(sqrt(k/m)*t)) = (k/m) * (A^2-(L2-L)^2) = ((F^2)/(k*m) - (k/m)*(L2-L)^2). Finally we have
v = sqrt((F^2)/(k*m) - (k/m)*(L2-L)^2).

[Ether]

Good stuff.

In post #2, lemiant used energy conservation to find the solution*.

In post #6, dan2915 used the equations for simple harmonic motion.

Let me add a third way: use Newton's 2^nd Law:

F = -k*x = M*a = M*(dV/dt) = M*(dx/dt)(dV/dx) = M*V*(dV/dx)

=> M*V*(dV/dx) = -k*x

=> V*dV = -(k/M)*x*dx

Integrate the left side from V=0 to V=V₂, and the right side from x=-F/k to x=(L-L₂), and solve for V₂

This Quiz was inspired by post #5 in this thread.


*neglecting the error in the final step

[Ether]

Originally Posted by lemiant

Energy in the system = F^2 / 2K

Did you mean F^2/(2K) ?