Question

A body of mass $m$ attached to the spring experiences a drag force proportional to its velocity and an external force $F\left(t\right)=F_o\cos {\omega }_{o}t$. The position of the mass at any point in time can be given by:

• A. $x\left(t\right)=c_1\sin (\omega t+\varphi )+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$
• B. $x\left(t\right)=c_1\cos (\omega t+\varphi )+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$
• C. $x\left(t\right)=c_1\sin \left(\omega t\right)+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$
• D. $x\left(t\right)=c_1\cos \left(\omega t\right)+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $x\left(t\right)=c_1\cos (\omega t+\varphi )+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$
B $x\left(t\right)=c_1\sin (\omega t+\varphi )+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$
C $x\left(t\right)=c_1\sin \left(\omega t\right)+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$
D $x\left(t\right)=c_1\cos \left(\omega t\right)+\left(\frac{F_o}{{\omega }^2-{\omega }_o^2}\right)\cos {\omega }_{o}t$

Initially, we already know that the displacement at any moment (Instant) of time for spring mass system is:

$x=a\sin (\omega t+\varphi )$ or,

$x=a\cos (\omega t+\varphi )$

And here an external force is experienced thus, all the four options give position at any time.