Differentiaion, Finding maximum points, and where x cuts axis PLS HELP!!!!!?

Question

A system of compression springs are compressed and let free to oscillate and found to give a displacement x:

x=4cos10pt+3sin10pt

p=3.147(pie)

Where t is the time in seconds find
(a)Oscillation frequency
(b)Maximum displacement
(c)Time when the displacement is maximum
(d)Time when the displacement is zero

workings if possible so i can see exactly how its done. cheers

Answer 1

answer(d):

put x=0 and work out the time

4cos10pt + 3sin10pt = 0
4cos10pt = -3sin10pt
1 = -3sin10pt / 4cos10pt
1 = (-3/4)tan10pt

taking arctan (tan-inverse) find t.

answer(c):

take differentiation taking derivative of trigonometric terms and then of (10pt)

dx/dt = 4(10p)(-sin10pt) + 3(10p)(cos10pt) ------eq.1

place this equal to zero and find t.

0 = -40psin10pt + 30pcos10pt
40psin10pt = 30pcos10pt
40psin10pt/30pcost10pt = 1
(4/3) tan10pt = 1

find t.

NOTE: you will get two answers one for mawimum point and other for minimum. (As spring will move to and fro and will attain maximum distance on either side so displacement will be positive for one and negative for one as this is a vector).

technically both answers are correct, but if you regard one as positive and other as negative, the to find out which one is positive, differentiate again eq 1 and put both values of t one by one. the value which gives NEGATIVE result is the answer.

answer(b):

put this t which you have found in part c in the original equation to get the value. (By the way if you put both values of t, you will get same result, one positive and one negative.)

answer(a):
i don't know if your level is high to accespt this answer, if not, my apologies. the correct answer is 10p. but exactly how is this done is

form this equation in terms of x = cos (10pt-b)

let 4 = cos b
and 3 = sin b

putting in equation

x = cosb x cos10pt + sinb x sin10pt

this makes

x = cos (10pt-b)

where 10p is the oscillating frequency and b is the initial phase. this can be cos(10pt+b) if you put 3 = -sinb
why negative b and positive b?

this is because if you take the direction of compression positive then it is cos(10pt +b) and the cos wave will shift to left (in graph) and taking direction of compression negative, it will be cos(10pt-b) maikng graph shift right. This b is the value of x (displacement) at t=0 (the y intercept)i.e. the point to which the spring is compressed before freed.

Answer 2

a) w=2pi f = 10 pi so f = 5 hz
You can always write
4 cos wt +3 sin wt as A cos(wt-phi)=Acoswt*cos phi+Asinwt*sin phi so
Acos phi = 4
A sin phi =3
so squaring and summing
A^2=25 and A=5
tan phi =3/4 so phi = 0.6435 rad

x=5cos(10pi t-0.6435)
So the maximum displacement is 5
cos (10pi t-0,6435)=1
10pi*t-0.6435 =2kpi so t = 0.6435/10pi +k/5(k any integer)
d) cos(10 pi t-0.6435)=0
10 pi*t-0.6435 = pi/2+kpi
t= 1/20+k/10 +0.6435/10pi