Question

An object of mass m is connected to a light spring with a force constant of kH N/meter which oscillates on a frictionless horizontal surface with Simple Harmonic Motion. At t = 0 the spring was at rest but is compressed x = A meter maximum during oscillation. Write the equation of motion from Newton's 2nd law FH = m·a and Hook's Law FH = -kH·x. Because of the starting position assume a solution is x = A sin(ωt) a sinusoidal function of time [NOTE it would be a cosine if the spring was compressed initially]. What is the velocity v (velocity) and a (acceleration) using the relations we derived in class. Plug these into the equation of motion and find the value of ω using the symbols kH and m.

HINT

a) with Hook's Law Force write the equation of motion from Newton's 2nd Law: -kH·x =

From initial conditions as noted above ... Assume a solution to this equation is x = A·sin(ω·t)

b) write the equations for v = [express algebraically with A, ω, t, trig function]

c) and an equation for for a = REMEMBER ω is a Greek letter

d) Put the answer for c) into answer a) and derive ω =

e) Also show that velocity can be found to be a sinusoidal function of time using energy relationships.

Hint: Start with the equation for energy: WH + Wk = Wtotal where ΔW = F•Δx

WH =

1

2

kH x2 potential energy and WK =

1

2

m v2 kenetic energy, find total at x=A, v=0 then solve

v = ±

kH

m

· (A2 − x2)

and then put in the solution for x assumed and ω, simplify with algebra and trig

derive v = Check your work, compare to the answer for b)

Describe how you did it [and show your work]

Answer 1