If anyone can help, it would be much appreciated. Consider a block of mass M tha
ID: 2122859 • Letter: I
Question
If anyone can help, it would be much appreciated.
Consider a block of mass M that sits atop 2 solid disks of raduis r and mass m. On the top of block sits a smaller block of mass mu attached to a spring with spring constant k. See picture below. There is no friction between the blocks. Initially the system is at rest, then at t=0 the block is given an initial velocity v0 with respect to the ground and displacement x0 with respect to the cart (so the spring is compressed x0). Note that the large block of mass M is not attached to the disks, but you may assume that the large block of mass M does not fall off of the solid disks. During one complete cycle, how far does the large block (of mass M) move? What is the maximum velocity of the large block (of mass M)? Write the velocity of the small block of mass mu as the velocity of the small cart with respect to the large cart (V big cart - small cart ) plus the large block with respect to the ground. Take the time derivative of the energy conservation equation and the momentum conservation equation. If you let x represent the displacement from equilibrium of the spring V big cart - small cart = dx/dt. Combine these results to get an equation that looks like d2x/dt2 = (something) x. As you have (or will shortly) discuss in class, that "(something)" is the frequency squared of oscillation, omega . Find the frequency of oscillation. Hints/Advice: Think about how far the large block moves when the center of mass of the wheels travels adistance d. For parts 1 and 2 momentum, energy, and center of mass all play a role here, so you'll want to pick a coordinate system on the ground to measure the positions and velocities. Extra Credit - Due 8/1 at beginning of class: Consider a particle of mass m traveling in a straight line with velocity v, v>0, along the line (x,b) i e the horizontal line with y=b. Assume that the mass starts at x=infinity. Is angular momentum conserved? If so write the angular momentum in terms of constants of the problem, if not write an expression for the angular momentum as a function of time and identify the torque.Explanation / Answer
the kinetic energy of the spring is
K = (1/2)k x d^2
the kinetic energy of the mass u is
K1 = (1/2)u x v^2
here,K = K1
or u x v^2 = k x d^2
or v = (k/m)^1/2 x d
where v is speed of mass u
the maximum velocity of mass M is
v_max = d x w = d x (M/k)^1/2
the kinetic energy of the disk is
K2 = I x w^2
where I is moment of inertia and w is angular frequency
or w = (K2/I)^1/2
here,K2 = (1/2)M x v_max^2 and I = (m x R^2/2)