A 5.55 kg block covered in sandpaper is pushed along the ceiling of a room under construction. The block is pushed across the ceiling with a force of 77.9 N directed at an angle of 71.5° to the horizontal. If the coefficient of kinetic friction between the paper and the ceiling is 0.574, what acceleration does the block undergo?
The key to solving this kind of problems is to find the net force acting on the block, free of weight (mg) and frictional forces.
Consider forces acting on the block along the horizontal and vertical directions.
Let
force applied on the block, F = 77.9 N
angle with horizontal, θ = 71.5°
Coeff. of kinetic friction, μk=0.574
mass of block, m = 5.55 kg
acceleration due to gravity, g = 9.80
Normal reaction on the block from the ceiling, N
Let's sum forces in the vertical direction first:
upward component of force - weight = 0
Fsinθ - mg - N= 0
from which N can be found.
Consider horizontal forces, positive in direction of F.
net horizontal force, H = Fcosθ - μN
Using Force = mass * acceleration, we conclude that the horizontal acceleration of the block is given by
H = ma
where a= acceleration in the direction of H.
Can you substitute the numbers and work out the answer?
77.9Nsin71.5 - (5.55kg)(9.8m/s^2) - N = 0
N = 19.5N
H = 77.9cos71.5 - (0.574)(19.5N)= 13.5N
a = 13.53 N/5.55 kg = 2.44 m/s^2
Correct!
To find the acceleration of the block, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration:
F_net = m * a
In this case, the net force can be calculated by subtracting the force of friction from the applied force:
F_net = F_applied - F_friction
The force of friction can be found using the formula:
F_friction = μ * F_normal
where μ is the coefficient of kinetic friction and F_normal is the normal force. Since the block is being pushed upwards on the ceiling, the normal force is equal to the weight of the block, which can be calculated as:
F_normal = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, let's calculate the different components:
F_normal = (5.55 kg) * (9.8 m/s^2) = 54.39 N
F_friction = (0.574) * (54.39 N) = 31.20 N
F_net = (77.9 N) - (31.20 N) = 46.7 N
Finally, we can calculate the acceleration of the block by rearranging Newton's second law equation:
a = F_net / m = (46.7 N) / (5.55 kg) = 8.41 m/s^2
Therefore, the block undergoes an acceleration of 8.41 m/s^2.