A 3.60 kg block is pushed along the ceiling with an constant applied force of F = 75.5 N that acts at an angle θ = 61° with the horizontal, as in the figure below. The block accelerates to the right at 5.76 m/s2. Determine the coefficient of kinetic friction between the block and the ceiling.
To determine the coefficient of kinetic friction between the block and the ceiling, we need to analyze the forces acting on the block.
First, let’s identify the forces involved:
1) The applied force (F): This is the force exerted on the block, which is given as 75.5 N at an angle of 61° with the horizontal.
2) The force of gravity (mg): This is the weight of the block, which is given by the mass (m) multiplied by the acceleration due to gravity (g). Since we’re dealing with a block pushed along the ceiling, the force of gravity acts vertically downward.
3) The normal force (N): This is the force exerted by the ceiling on the block, perpendicular to the surface of contact. In this case, it acts vertically upward.
4) The force of kinetic friction (fk): This is the force opposing the motion of the block, acting horizontally in the opposite direction to the applied force.
Now, let’s break down the forces along the x and y axes:
Along the x-axis:
- The applied force (F) can be resolved into its horizontal component (Fh) and vertical component (Fv) using trigonometry.
- The force of kinetic friction (fk) is the only force acting horizontally and opposing the motion.
- Since the block accelerates to the right, the net force along the x-axis is given by the equation: F - fk = m * ax (where ax is the acceleration along the x-axis).
Along the y-axis:
- The force of gravity (mg) and the vertical component of the applied force (Fv) cancel each other out.
- The normal force (N) equals the force of gravity (mg).
Now, let’s calculate the forces involved:
1) Resolve the applied force into its horizontal (Fh) and vertical (Fv) components:
Fh = F * cos(θ) = 75.5 N * cos(61°)
Fv = F * sin(θ) = 75.5 N * sin(61°)
2) Calculate the net force along the x-axis using:
F - fk = m * ax
(75.5 N * cos(61°)) - fk = (3.60 kg) * (5.76 m/s²)
3) Calculate the normal force (N) using:
N = mg
N = (3.60 kg) * (9.8 m/s²)
4) Rearrange the equation from step 2) to solve for fk:
fk = (75.5 N * cos(61°)) - (3.60 kg * 5.76 m/s²)
Finally, to find the coefficient of kinetic friction (μk), we can use the equation:
μk = fk / N
Substitute the values of fk and N into the equation to get the coefficient of kinetic friction.
First get the net force from the acceleration.
Fnet = M*a = 3.6*5.76 = 20.74 N
The net force is the applied horizongtal force component MINUS the friction force.
20.74 = 75.5 cos61 - (75.5sin61-Mg)*U
= 36.60 - (66.03 - 35.28)*U
= 36.60 - 30.75U
Solve for the friction coefficient, U