Question

1. To move a large crate across a rough floor, you push on it with a force F at an angle of 21° below the horizontal, as shown in Figure 6–18. Find the force necessary to start the crate moving, given that the mass of the crate is 32 kg and the coefficient of static friction between the crate and the floor is 0.57.
A). Find the acceleration of the crate if the applied force is 330 N and the coefficient of kinetic friction is 0.45.

Answer 1

Given that the mass of the crate is 32 kg and the coefficient of static friction between the crate and the floor is 0.57, the horizontal force required to start the crate moving is

$$F_H=F\text{ cos}21°=\mu N,\\ N=mg+F\text{ sin}21°,\\ F\text{ cos}21°=\mu mg+F\text{ sin}21°,\\\space\\ F=\frac{\mu mg}{\text{cos}21°-\text{sin}21°}=311\text{ N}.$$

The acceleration of the crate if the applied force is 330 N and the coefficient of kinetic friction is 0.45 can be determined from the following condition by Newton's second law:

$$ma=F\text{ cos}21°-\mu N,\\ N=mg+F\text{ sin}21°,\\ ma=F\text{ cos}21°-\mu(mg+F\text{ sin}21°),\\\space\\ a=\frac{1}{m}[F\text{ cos}21°-\mu(mg+F\text{ sin}21°)]=7.5\text{ m/s}^2.$$