A 3.40 kg block is pushed along the ceiling with an constant applied force of F = 82.0 N that acts at an angle θ = 50° with the horizontal, as in the figure below. The block accelerates to the right at 5.80 m/s2. Determine the coefficient of kinetic friction between the block and the ceiling. μk =
To determine the coefficient of kinetic friction between the block and the ceiling, we need to use Newton's second law of motion and the concept of friction.
Let's break down the forces acting on the block:
1. The applied force (F): This force is acting in an upward direction at an angle of 50° with the horizontal.
2. The force of gravity (mg): This force is acting straight downward and is given by the mass (m) of the block multiplied by the acceleration due to gravity (g), which is approximately 9.8 m/s^2.
3. The normal force (N): This force is exerted by the ceiling and acts perpendicular to the surface. It cancels out the vertical component of the applied force and is equal in magnitude to the force of gravity (mg).
4. The force of kinetic friction (fk): This force acts in the opposite direction of the applied force. It can be calculated as the coefficient of kinetic friction (μk) multiplied by the normal force (N).
Now let's set up equations using Newton's second law:
1. In the horizontal direction:
Sum of forces in the x-direction = ma (acceleration in the x-direction = 5.80 m/s^2)
F*cos(θ) - fk = ma
2. In the vertical direction:
Sum of forces in the y-direction = 0 (since there is no vertical acceleration)
F*sin(θ) - mg + N = 0
Solving equation 2 for N:
N = mg - F*sin(θ)
Substituting this value for N in equation 1:
F*cos(θ) - fk = ma
F*cos(θ) - μk*(mg - F*sin(θ)) = ma
Now we can rearrange the equation to solve for the coefficient of kinetic friction (μk):
μk = (F*cos(θ) - ma) / (mg - F*sin(θ))
Substituting the given values:
F = 82.0 N
θ = 50°
m = 3.40 kg
a = 5.80 m/s^2
g = 9.8 m/s^2
μk = (82.0*cos(50°) - 3.40*5.80) / (3.40*9.8 - 82.0*sin(50°))
Now you can calculate the coefficient of kinetic friction (μk).
Use F = m*a and solve for the friction coefficient, which will be part of the Force term.
[82cos50 - (82sin50-3.40*g)*ìk]
= 3.40*5.8
The term in brackets [ ] is the net force along the ceiling.