We are given a box that slides up a ramp. To determine the force of friction we will use the following relationship:
[tex]F_f=\mu N[/tex]
Where.
[tex]\begin{gathered} N=\text{ normal force} \\ \mu=\text{ coefficient of friction} \end{gathered}[/tex]
To determine the Normal force we will add the forces in the direction perpendicular to the ramp, we will call this direction the y-direction as shown in the following diagram:
In the diagram we have:
[tex]\begin{gathered} m=\text{ mass}_{} \\ g=\text{ acceleration of gravity} \\ mg=\text{ weight} \\ mg_y=y-\text{component of the weight. } \end{gathered}[/tex]
Adding the forces in the y-direction we get:
[tex]\Sigma F_y=N-mg_y[/tex]
Since there is no movement in the y-direction the sum of forces must be equal to zero:
[tex]N-mg_y=0[/tex]
Now we solve for the normal force:
[tex]N=mg_y[/tex]
To determine the y-component of the weight we will use the trigonometric function cosine:
[tex]\cos 40=\frac{mg_y}{mg}[/tex]
Now we multiply both sides by "mg":
[tex]mg\cos 40=mg_y[/tex]
Now we substitute this value in the expression for the normal force:
[tex]N=mg\cos 40[/tex]
Now we substitute this in the expression for the friction force:
[tex]F_f=\mu mg\cos 40[/tex]
Now we substitute the given values:
[tex]F_f=(0.2)(10\operatorname{kg})(9.8\frac{m}{s^2})\cos 40[/tex]
Solving the operations:
[tex]F_f=15.01N[/tex]
Therefore, the force of friction is 15.01 Newtons.
