Q41E - University Physics with Modern Physics

Question

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 ${\mathit{\mu }}_{\mathbf{s}}$ .

Step-by-Step Solution

Answer

(a) The magnitude of force is $\frac{{\mu }_{k}mg}{\cos \theta -{\mu }_k\sin \theta }$ .

(b)  The critical value of ${\mu }_s$  is $cot\theta$ .

Step 1: Identification of given data:

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 $\theta$ .

Step 2: Significance of magnitude of force:

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.

Step 3: Determination of magnitude of force:

                       

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 $F=\frac{{\mu }_{k}mg}{\cos \theta -{\mu }_k\sin \theta }$ .

Step 4: Determination of critical value for μ s .

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 $cot\theta$ .