In an engine, a piston oscillates with simpleharmonic motion so that its positio

Question

In an engine, a piston oscillates with simpleharmonic motion so that its position varies according to thefollowing expression, where x is incentimeters and t is inseconds. x = (6.00 cm)cos(3t + /4) (a) At t = 0, find theposition of the piston.
___ cm

(b) What is its velocity?
___ cm/s

(c) What is its acceleration?
___ cm/s2

(d) Find the period and amplitude of themotion.
___ s
___ cm x = (6.00 cm)cos(3t + /4) (a) At t = 0, find theposition of the piston.
___ cm

(b) What is its velocity?
___ cm/s

(c) What is its acceleration?
___ cm/s2

(d) Find the period and amplitude of themotion.
___ s
___ cm

Explanation / Answer

Given      position     x = (6 cm ) cos ( 3t + /4 )    a) at t = 0             x =   (6.00cm )   cos ( 3(0) + / 4   )               = 6 cos ( / 4 )                = 4.24 cm b) velocity           v = dx / dt                =   d / dt ( (6 cm ) cos ( 3t + / 4 ))                = 6 sin (3t + / 4 ) *3                = 18 sin (3t + / 4 )   cm / s c )   acceleration           a   = dv / dt                    = d / dt ( 18 sin (3t + / 4 ) )                   =   54 cos( 3t + / 4 ) d) Given   position is compared tothe   general wave equation                y = A sin (t + )           Amplitude   A    = 6 cm           angular frequency   = 3                     time period   T = 2   /                                                  = 2 / 3   = 2.094 sec

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