A 0.25 kg block oscillates back and forth along a straight line on a frictionles

Question

A 0.25 kg block oscillates back and forth along a straight line on a frictionless horizontal surface. Its displacement from the origin is given by
x = (18 cm)cos[(11 rad/s)t + /2 rad]
(a) What is the oscillation frequency? (b) What is the maximum speed acquired by the block? (c) At what value of x does this occur? (d) What is the magnitude of the maximum acceleration of the block? (e) At what positive value of x does this occur? (f) What force, applied to the block by the spring, results in the given oscillation?

Explanation / Answer

part a )

w = 11 rad/s

w = 2pif

f = w/2pi

f = 1.75 Hz

part b )

dx/dt = v

v = -11 * 18 (cm) sin(11 t + pi/2 )

vmax = 11 * 18 *10^-2 = 1.98 m/s

part c )

sin (11t + pi/2 ) = +-1

11t + pi/2 = +-pi/2

x = 18 cos ( +-pi/2 ) = 0

x = 0 cm

part d )

a = dv/dt = -11^2 * 18(cm) *cos[11 t + pi/2 ]

max a = aw^2 = 18 *10^-2 x 11^2 = 21.78 m/s^2

for max = cos[11t + pi/2] = -1

x = 18 cm

part f )

F = -Kx

K = mw^2 = 0.25 * 11*11 = 30.25 N/m

F = -30.25 * x

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