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)
To find the mass of the box, we can use the equation that relates the force of friction to the normal force and coefficients of friction:
\(f_{\text{friction}} = \mu N\)
where \(f_{\text{friction}}\) is the force of friction, \(\mu\) is the coefficient of friction, and \(N\) is the normal force.
In this case, the normal force is equal to the weight of the box, since it is being held in place against the wall. The weight of an object is given by the equation:
\(W = mg\)
where \(W\) is the weight, \(m\) is the mass, and \(g\) is the acceleration due to gravity.
The force applied to the box can be resolved into its vertical and horizontal components. The vertical component of the force should balance the weight of the box, while the horizontal component provides the force of friction.
Let's calculate the mass of the box step-by-step:
Step 1: Calculate the normal force
Since the weight of the box is equal to the normal force, we have:
\(N = mg\)
Step 2: Calculate the vertical component of the applied force
The vertical component of the applied force is given by:
\(F_{\text{vertical}} = F_{\text{applied}} \cdot \sin(\theta)\)
where \(F_{\text{applied}}\) is the magnitude of the applied force and \(\theta\) is the angle above the horizontal. In this case, \(\theta = 29.8°\).
Step 3: Equate the vertical component of the applied force to the weight
Since the vertical component of the applied force should balance the weight, we have:
\(F_{\text{vertical}} = mg\)
Step 4: Calculate the force of friction
The force of friction is given by:
\(f_{\text{friction}} = \mu N\)
Step 5: Equate the force of friction to the horizontal component of the applied force
Since the box slides down unless the applied force has a magnitude of 11.1 N, we have:
\(f_{\text{friction}} = F_{\text{horizontal}} = F_{\text{applied}} \cdot \cos(\theta)\)
Step 6: Solve for the unknown mass
By combining Step 3, Step 4, and Step 5, we can solve for the mass of the box:
\(mg = \mu N = \mu (F_{\text{applied}} \cdot \cos(\theta)) \)
Rearranging the equation and solving for \(m\), we get:
\(m = \frac{\mu F_{\text{applied}} \cdot \cos(\theta)}{g}\)
Now, let's plug in the given values and calculate the mass of the box.
Given information:
\(\mu = 0.400\)
\(\mu_k = 0.320\)
\(F_{\text{applied}} = 11.1 \, \text{N}\)
\(\theta = 29.8°\)
\(g = 9.8 \, \text{m/s}^2\)
Calculating the mass of the box:
\( m = \frac{\mu F_{\text{applied}} \cdot \cos(\theta)}{g} \)
\( = \frac{0.400 \cdot 11.1 \, \text{N} \cdot \cos(29.8°)}{9.8 \, \text{m/s}^2} \)
\( ≈ 1.48 \, \text{kg}\)
Therefore, the mass of the box is approximately 1.48 kg.
To find the mass of the box, we need to use the given information and apply Newton's laws of motion.
Let's break down the problem step by step:
1. Start by identifying the forces acting on the box. In this case, we have the force applied upward at an angle of 29.8° with respect to the horizontal and the force of friction acting downward.
2. Resolve the applied force into its horizontal and vertical components. The vertical component will help us determine the normal force, and the horizontal component will help us calculate the force of friction.
3. Calculate the vertical component of the applied force:
F_vertical = F_applied * sin(29.8°)
4. Determine the normal force on the box, which is equal to the weight of the box. The normal force is the force exerted by the wall perpendicular to the surface of contact with the box. It counterbalances the weight of the box.
N = m * g, where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s²).
5. Calculate the force of friction using the coefficient of static friction and the normal force:
F_friction = μ_static * N
6. Set up an equilibrium equation in the vertical direction:
ΣF_vertical = 0
Normal Force - F_vertical = 0
7. Solve for the normal force:
Normal Force = F_vertical = F_applied * sin(29.8°)
8. Use the coefficient of static friction and the normal force to calculate the maximum value of the force applied before the box starts sliding:
F_max = μ_static * Normal Force
9. Compare the maximum force of friction to the force applied to determine if the box slides or not. If the applied force is less than or equal to the maximum force of friction, the box will not slide. Otherwise, it will slide.
10. If the box does not slide, the applied force is equal to the force of friction. If the box slides, use the coefficient of kinetic friction and the normal force to calculate the force of friction when sliding:
F_friction = μ_kinetic * Normal Force
11. Finally, calculate the mass of the box using the force of friction:
F_friction = m * g
m = F_friction / g
Plug in the given values and follow these steps to find the mass of the box.