A small box is held in place against a rough wall by someone pushing on it with a force directed upward at 29.8° above the horizontal. The coefficients of static and kinetic friction between the box and wall are 0.400 and 0.320, respectively. The box slides down unless the applied force has maganitude 11.1 N. What is the mass of the box?

Question

A small box is held in place against a rough wall by someone pushing on it with a force directed upward at 29.8° above the horizontal. The coefficients of static and kinetic friction between the box and wall are 0.400 and 0.320, respectively. The box slides down unless the applied force has maganitude 11.1 N. What is the mass of the box?

(in kg)

Answer 1

To solve this problem, we need to analyze the forces acting on the box.

Let's break down the forces:
1. The force of gravity acting vertically downward.
2. The normal force acting perpendicular to the rough wall.
3. The pushing force applied at an angle above the horizontal.
4. The static friction force that prevents the box from sliding down when the applied force is less than 11.1 N.
5. The kinetic friction force that opposes the motion when the applied force is equal to or greater than 11.1 N.

First, let's calculate the components of the applied force:
Horizontal component: F_horizontal = F_applied * cos(29.8°)
Vertical component: F_vertical = F_applied * sin(29.8°)

The forces acting vertically are:
1. Force of gravity: F_gravity = m * g

The forces acting horizontally are:
1. Normal force: F_normal = F_gravity * cos(θ)
2. Static friction force: F_static = μ_static * F_normal

Note that the normal force is equal in magnitude and opposite in direction to the horizontal component of the force of gravity. In other words, F_normal = -F_gravity * sin(θ).

To find the mass of the box, we need to equate the static friction force to the maximum available static friction force, which prevents the box from sliding down. The maximum static friction force can be calculated as follows:
F_static_max = μ_static * F_normal

When the applied force is equal to or greater than 11.1 N, the box starts sliding down, and the force of static friction is replaced by kinetic friction.

So, we can equate the kinetic friction force to the applied force:
F_kinetic = μ_kinetic * F_normal

Finally, to solve for the mass of the box, we need to use the formula:
F_net = m * a

Since the box is not accelerating vertically, we can write:
F_net_vertical = F_gravity + F_vertical = 0

Therefore,
F_gravity = -F_vertical

We can now substitute all the equations and solve for the mass of the box.