Answer
$P=313.92N$
Work Step by Step
The required maximum force can be determined as follows:
We know that the equation of motion in x-direction is given as
$\Sigma F_x=ma_x$
$\implies -F_f=-ma$
$\implies \mu_s N=ma$
We plug in the known values to obtain:
$0.2N=150a$
$\implies N=750a$ [eq(1)]
Now, applying the equation of motion in y-direction
$\Sigma F_y=ma_y$
$\implies N-W=m(0)$
$\implies N-150\times 9.81=0$
This simplifies to:
$N=1471.5N$ [eq(2)]
From eq(2), we plug in this value in eq(1) to obtain:
$1471.5=750a$
$\implies a=1.962m/s^2$
The sum of moments about the point A is given as
$\Sigma M_A=\Sigma M_{kA}$
$\implies W(x)=ma(0.5)$
$\implies 150\times 9.81x=150\times 1.962\times 0.5$
This simplifies to:
$x=0.1m$
Now, we apply the equation of motion to figure (2) in x-direction
$\Sigma F_x=ma_x$
$\implies -P=-(m_{crate}+m_{cart})a$
We plug in the known values to obtain:
$P=(150+10)\times 1.962$
This simplifies to:
$P=313.92N$
