Question

A car begins to move at time $t=0$ and then accelerates along a straight track with a velocity given by $V\left(t\right)=2t^2{\text{ms}}^{–1}$ for $0\le t\le 2$, where $t$ is time in second. After the end of acceleration, the car continues to move at a constant speed. A small block initially at rest on the floor of the car begins to slip at $t=1\text{s}$ and stops slipping at $t=3\text{s}$. The coefficient of static and kinetic friction between the block and the floor are ${\mu }_s$ and ${\mu }_k$ respectively. Find the value of $\frac{3{\mu }_s}{{\mu }_k}$ ($g=10m{s}^{-2}$)

Answer 1

$V=2t^2$
$a=4t$
at $t=1\text{sec}$, slipping occur
$ma={\mu }_{s}mg$
$4t={\mu }_{s}g$
${\mu }_s=0.4$
$t=1\to t=3$ slipping occurs between $t=1$ & $t=3\text{sec}$
$(4t-{\mu }_{k}g)=\frac{d{V}_r}{dt}$
$V_r={[2t^2-{\mu }_{k}gt]}_1^2,V_r=2\times 3-{\mu }_{k}g$
$0=V_r-{\mu }_{k}gt$
$0=V_r-{\mu }_{k}g$
${\mu }_k=0.3,\frac{3{\mu }_s}{{\mu }_k}=4$