A block of mass attached to an ideal horizontal spring of force constant performs…

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Oscillations · Oscillations

Problem

A block of mass

m

attached to an ideal horizontal spring of force constant

k

performs simple harmonic motion on a frictionless surface. The block is released from rest at the extreme left position

x=-A

and subsequently oscillates between

x=-A

and

x=+A

. On a set of axes whose horizontal axis represents displacement

x

(from

-A

+A

) and whose vertical axis represents energy, three curves are to be sketched: the elastic potential energy

U(x)

, the kinetic energy

K(x)

, and the total mechanical energy

E_{\text{tot}}

of the block–spring system. (a) Sketch, on the same set of axes, the qualitative shapes of

U(x)

K(x)

, and

E_{\text{tot}}

. Clearly label the amplitude

A

on the horizontal axis and indicate the numerical value of

E_{\text{tot}}

on the vertical axis in terms of

k

and

A

. No numerical scale is required beyond these labels. (b) Describe how the individual values of

U

and

K

change as the block moves (i) from

x=-A

x=0

and (ii) from

x=0

x=+A

. Your answer should refer explicitly to the relative magnitudes of

U

and

K

at the endpoints and at the midpoint. (c) Explain why the curve representing

E_{\text{tot}}

appears as a horizontal line on the graph, and discuss what physical principle this illustrates for the motion of the block.