Question

A 0.57 kg block oscillates back and forth along a straight line on a frictionless horizontal surface. Its displacement from the origin is given by
x = (13 cm)cos[(20 rad/s)t + π/2 rad]
(a) What is the oscillation frequency (in Hz)? (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?
How do you even start this problem? Please help

Answer 1

first find the period. That is when the argument of the cos changes by 2 pi
when t = 0
20(0) + pi/2 = pi/2
2 pi + Pi/2 = 2 pi + pi/2
so
period when 20t = 2 pi
or T = 2 pi/20 = pi/10
f = 1/T = 10/pi

b
dx/dt = .13 (20)(-sin(20t+pi/2))
max when |sin| = 1
dx/dt max = 2.6 m/s

c
well at lots of values of x but one is when
20t+pi/2 = pi/2
or t = 0

d take the next derivative and repeat

e now you have the acceleration d^2x/dt^2
F = m a