Question

Consider the following four blocks, all of which are attached to identical springs:

block A: mass = 200·g, spring stretched 10·cm
block B: mass = 800·g, spring stretched 20·cm
block C: mass = 400·g, spring stretched 20·cm
block D: mass = 400·g, spring stretched 20·cm.

(a) All the blocks are placed on a level surface that has no significant friction. For each of the following ranking tasks use the symbols > and = to indicate your ranking (ties are possible), listing any that are equal in alphabetical order. For example, if Y and Z are the same and are both greater than X, then you would enter Y=Z>X (note: Z=Y>X would be incorrect, since Y and Z, being equal, must be in alphabetical order). Rank the blocks, from largest to smallest, based on the potential energy stored in the spring attached to each block.
largest PE  smallest PE

(b) The springs are released and the blocks begin to move. Rank the blocks, from largest to smallest, based on their kinetic energy when they reach the mid-point of their oscillation (i.e., when their springs are unstretched)?
largest KE  smallest KE

(c) Compare the relative speed of blocks A and B when they reach their mid-points. Choose one answer only.

speed of A > speed of Bspeed of A = speed of B    speed of B > speed of A

(d) Compare the relative speed of blocks A and D when they reach their mid-points. Choose one answer only.

speed of A > speed of Dspeed of A = speed of D    speed of D > speed of A

Answer 1

Given

block A: mass = 200·g, spring stretched 10·cm

block B: mass = 800·g, spring stretched 20·cm

block C: mass = 400·g, spring stretched 20·cm

block D: mass = 400·g, spring stretched 20·cm.

a)

elastic potential energy is U = 0.5*k*x^2

all are of same k value so U depends on x value  

from the given data B = C = D > A

b)

the largest kinetic energy will be at mean position  

by conservation of energy 0.5*k*x^2 = 0.5 *m*v^2

v = sqrt(k/m) (x)

k is constant ==> v = sqrt(1/m)(x) ==> k1 = 0.5*m1*(1/m1)x^2 = 0.5*x^2

means the kinetic energy depends on the stretch in the spring  

so the ranking is B = C = D > A

c) speed of block A when it reach the mid point

VA = sqrt(1/m)(x) = sqrt(1/0.2)(0.1) m/s = 0.22361 m/s

VB = sqrt(1/m)(x) = sqrt(1/0.8)(0.2) m/s = 0.22361 m/s

both having same speed when they reach the midpoint

speed of A = speed of B

d)

speed of block A when it reach the mid point

VA = sqrt(1/m)(x) = sqrt(1/0.2)(0.1) m/s = 0.22361 m/s

speed of block D when it reach the mid point

VD = sqrt(1/m)(x) = sqrt(1/0.4)(0.2) m/s = 0.316 m/s

speed of D > speed of A