Answer:
161.7 N
Explanation:
To find the magnitude of the horizontal force required to start sliding a 55.0-kg box on a horizontal surface with a coefficient of static friction of 0.300, we can use the following steps:
Step 1: Calculate the normal force (n)
In this case we can sum the forces acting vertically on the box to find the normal force:
[tex]\sum \vec F_y: \vec n -\vec w=0[/tex]
[tex]\boxed{ \left \begin{array}{ccc} \text{\underline{Weight of an Object:}} \\\\ \vec w = mg \\\\ \text{Where:} \\ \bullet \ \vec w \ \text{is the weight of the object (force due to gravity)} \\ \bullet \ m \ \text{is the mass of the object} \\ \bullet \ g \ \text{is the acceleration due to gravity} \end{array} \right.}[/tex]
[tex]\Longrightarrow \vec n = mg\\\\ \\\\\Longrightarrow \vec n = (55.0 \text{ kg})(9.8 \text{ m/s}^2)\\\\\\\\\therefore \vec n = 539 \text{ N}[/tex]
Step 2: Calculate the maximum static friction force (f_s)
The static friction force can be calculated using the coefficient of static friction (μ_s) and the normal force (N):
[tex]\boxed{ \begin{array}{ccc} \text{\underline{Formula for Static Friction:}} \\\\ \vec f_s \leq \mu_s \vec n \ \Big(\text{Note: } \vec f_{s_{\text{MAX}}} = \mu_s \vec n\Big) \\\\ \text{Where:} \\ \bullet \ \vecf_s \ \text{is the actual static frictional force} \\ \bullet \ \mu_s \ \text{is the coefficient of static friction} \\ \bullet \ \vec n \ \text{is the normal force} \end{array}}[/tex]
[tex]\Longrightarrow \vec f_s = (0.300)(539 \text{ N})\\\\\\\\\therefore \vec f_s = 161.7 \text{ N}[/tex]
Step 3: Determine the horizontal force required.
Summing the forces horizontally we find,
[tex]\sum \vec F_x: \vec F_\text{min} -161.7 \text{ N}=0\\\\\\\\\therefore\boxed{ \vec F_\text{min} = 161.7 \text{ N}}[/tex]
The magnitude of the horizontal force required to start sliding the box is equal to the maximum static friction force.
