A small box is held in place against a rough vertical wall by someone pushing on it with a force directed upward at 28∘ above the horizontal. The coefficients of static and kinetic friction between the box and wall are 0.40 and 0.30, respectively. The box slides down unless the applied force has magnitude 41N .What is the mass of the box in grams?
To solve this problem, we need to consider the forces acting on the box. Let's break it down step by step:
1. The force applied to the box by the person has two components: a vertical component and a horizontal component. The vertical component opposes the gravitational force acting on the box, while the horizontal component causes friction.
2. The gravitational force acting on the box can be calculated using the formula F = mg, where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. The coefficient of static friction (μs) is the maximum friction that can be experienced between two surfaces without causing motion. The formula to calculate the maximum static friction force (Ffriction) is Ffriction = μs * N, where N is the normal force.
4. The normal force (N) is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is equal to the gravitational force acting on the box.
5. For the box to remain stationary, the horizontal component of the applied force should be equal to the static friction force. Therefore, we can set up an equation: Fhorizontal = Ffriction.
6. The kinetic friction force (Ffriction-kinetic) is the force opposing the motion between two surfaces in contact. It can be calculated using the formula Ffriction-kinetic = μk * N, where μk is the coefficient of kinetic friction.
7. Once the box starts sliding, the force applied by the person becomes the kinetic friction force.
Now, let's apply these steps to find the mass of the box:
1. Calculate the gravitational force acting on the box:
Fgravity = mg
2. Calculate the normal force:
N = Fgravity
3. Calculate the maximum static friction force:
Ffriction = μs * N
4. Set up an equation for the horizontal component of the applied force:
Fhorizontal = Ffriction
5. Calculate the kinetic friction force:
Ffriction-kinetic = μk * N
6. Use the fact that the applied force has magnitude 41N for the box to remain stationary, so the horizontal component of the applied force should be equal to the maximum static friction force:
Fhorizontal = 41N
7. Substitute the values into equations and solve for mass:
Fhorizontal = Ffriction
41N = μs * N
41N = μs * Fgravity
41N = 0.40 * mg
m = (41N) / (0.40 * g)
8. Convert the mass to grams by multiplying it by 1000.
Following these steps, you will get the mass of the box in grams.