A 0.23 kg block oscillates back and forth along a straight line on a frictionles

Question

A 0.23 kg block oscillates back and forth along a straight line on a frictionless horizontal surface. Its displacement from the origin is given by the following equation. x = (16 cm)cos[(11 rad/s)t + pi/2 rad] (a) What is the oscillation frequency? Hz (b) What is the maximum speed acquired by the block? cm/s (c) At what value of x does this occur? cm (d) What is the magnitude of the maximum acceleration of the block? cm/s^2 (e)At what value of x does this occur? cm (f) What force, applied to the block by the spring, results in the given oscillation? (Give your answer in terms of x.) Vector F = N/cm

Explanation / Answer

a) f= 11/2pi Hz

x=16cos(11t +pi/2) ------>(1)

b) dx/dt = -(16sin(11t + pi/2))(11) =V ----> (2)

dv/dt = -16cos(11t + pi/2)(121)=0 -----> (3)

11t + pi/2 = pi

t = pi/11 - Substitute to (2)

vmax = 176cm/s

c) substitute pi/11 to (1)

x = 0

d) differentiating (3)

we get d2v/dt2 = 21,296sin(11t + pi/2) = 0

t = pi/22 sec

subst. to (3)

amax = 1936cm/s2

e)

xamax = 16cos((pi/22)(11) + pi/2)

= 16cm

f) F = kx

w=(k/m)0.5

11=(k/0.23)0.5

k= 27.83

F=(27.83)(0.16)

= 4.45N

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