SHM question (1 Viewer)

[dingodog]

May someone please show me working out for this question?

a) If x=asin(nt) + bcos(nt), find the acceleration of the particle in terms of t and show that v=-n^2.x (completed myself)
ans:
x = asin(nt) + bcos(nt)
v = ancos(nt) - bnsin(nt)
a = -n^2(asin(nt) + bcos(nt)) = -n^2.x

b) Find the amplitude and period of motion (please give reasons why also)
ans: square root(a^2+b^2)

c) Find the maximum velocity (please give reasons why also)
ans: n.square root(a^2 + b^2)

 

[rand_althor]

b) Find the amplitude and period of motion (please give reasons why also)
ans: square root(a^2+b^2)

c) Find the maximum velocity (please give reasons why also)
ans: n.square root(a^2 + b^2)

For both of these, you can combine the terms in displacement and velocity involving acos(nt) and bsin(nt) using the auxiliary angle method, to get the functions in a form such as Rsin(nt+c), where R=sqrt(a^2+b^2). The R term is the only one influencing the amplitude in this case. So the displacement function will have a maximum of sqrt(a^2+b^2) and a minimum of -sqrt(a^2+b^2), and the velocity function will have a maximum of n*sqrt(a^2+b^2) and a minimum of -n*sqrt(a^2+b^2) (you factor out the n for the velocity function before using the auxiliary angle method).

 

[dingodog]

Thanks man