Can someone help me answer this SHM problem?

The energy equation for this is:

• #E = T + U#, where:

• #{(T = "kinetic energy" = 1/2 mv^2),(U = "potential (spring) energy" = 1/2 k x^2 ),(E = "total energy of the block" = 1/2 mv^2 + 1/2 k x^2):}#,

• ...in each case, at time #t#.

#E# is constant through time, so the SHM involves energy switching periodically between #T# and #U#.

At maximum extension, #x_("max")#:

• #v = 0, qquad E = U_("max") = 1/2 kx_("max")^2#

At zero extension, the equilibrium position:

• #x = 0, qquad E = T_("max") = 1/2 mv_("max")^2#

Finally, using energy conservation again:

• #E = 1/2 kx_("max")^2 = 1/2 mv_("max")^2 = " const"#

Now ANSWERING the questions:

A. energy of the system?

Because #E = 1/2 kx_("max")^2#, then #E propto x_("max")^2#.

Energy will quadruple.

B. maximum velocity of the oscillating mass?

#1/2 kx_("max")^2 = 1/2 mv_("max")^2 implies abs (v_("max") )propto abs ( x_("max"))#

Max velocity will double

C. acceleration of the mass?

For the spring:

• #F = - kx#

From Newton's 2nd Law, for the mass:

• #bbF = mbba qquad = - kbbx#

#:. a = - k/m x, qquad implies abs(a_"max") propto abs(x_("max")#

Acceleration through the equilibrium will still be zero, but the magnitude of acceleration at the extremities of the motion, when the mass is changing direction, will double