Question

Determine the velocity and position as a function of time for the time force F(t)=F Cos^2(WT). Generate plots for the resulting equation.

Answer 1

Answer with Explanation:

We know from newton's second law that acceleration produced by a force 'F' in a body of mass 'm' is given by

`a=(Force)/(Mass)=\farc{F}{m}`

In the given case the acceleration equals

`a=(F_(o)cos^(2)(\omega t))/(m)`

Now by definition of acceleration we have

`a=(dv)/(dt)\\\\\int dv=\int adt\\\\v=\int adt\\\\v=\int (F_(o)cos^(2)(\omega t))/(m)\cdot dt\\\\v=(F_(o))/(m)\int cos^(2)(\omega t)dt\\\\v=(F_(o))/(\omega m)\cdot ((\omega t)/(2)+(sin(2\omega t))/(4)+c)`

Similarly by definition of position we have

`v=(dx)/(dt)\\\\\int dx=\int vdt\\\\x=\int vdt`

Upon further solving we get

`x=\int [(F_(o))/(\omega m)\cdot ((\omega t)/(2)+(sin(2\omega t))/(4)+c)]dt\\\\x=(F_(o))/(\omega m)\cdot ((\int (\omega t)/(2)+(sin(2\omega t))/(4)+c)dt)\\\\x(t)=(F_(o))/(\omega m)\cdot ((\omega t^(2))/(4)-(cos(2\omega  t))/(8\omega )+ct+d)`

The plots can be obtained depending upon the values of the constants.