A large crate with mass rests on a horizontal floor. The coefficients of friction between the crate and the floor are ${\mathit{\mu }}_{\mathbf{s}}$and ${\mathit{\mu }}_{\mathbf{k}}$. A woman pushes downward with a force $\overrightarrow{\mathbf{F}}$on the crate at an angle $\mathit{\theta }$below the horizontal. (a) What magnitude of force $\overrightarrow{\mathbf{F}}$is required to keep the crate moving at constant velocity? (b) If ${\mathit{\mu }}_{\mathbf{s}}$is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. Calculate this critical value of ${\mathit{\mu }}_{\mathbf{s}}$.

The given data can be listed as below.

• The coefficient of static friction is ${\mu }_s$.
• The coefficient of kinetic friction is ${\mu }_k$.
• Mass of the crate is m .
• The angle of crate with horizontal is $\theta$.

The magnitude of force refer to when all the forces act on an object at a same time. If forces applied on an object are acting in a same direction than the magnitude of the force increases. If forces act in different directions then the magnitude of force will decrease.

From the free body diagram the frictional force is expressed as,

$F_f={\mu }_{k}N$

Here, ${\mu }_k$is the coefficient of kinetic friction, N is the normal force.

Substitute $F\cos \theta$for $F_f$and $F\sin \theta +mg$for N in the above equation.

$$\begin{gathered} F\cos \theta ={\mu }_k\left(F\sin \theta +mg\right) \\ F=\frac{{\mu }_{k}mg}{\cos \theta -{\mu }_k\sin \theta } \end{gathered}$$

Hence, required magnitude of force is $F=\frac{{\mu }_{k}mg}{\cos \theta -{\mu }_k\sin \theta }$.

From the free body diagram the coefficient of static friction expressed as,

${\mu }_s=\frac{F_f}{N}$

Here, ${\mu }_s$is the coefficient of static friction, $F_f$is the frictional force and N is the normal force.

Substitute $F\cos \theta$for $F_f$, $F\sin \theta$ for N in the above equation.

$$\begin{gathered} {\mu }_s=\frac{F\cos \theta }{F\sin \theta } \\ =cot\theta \end{gathered}$$

Hence, the required critical value for static friction is $cot\theta$.