A mop is pushed across the floor with a force F of 36.0 N at an angle of Θ = 50.0° (see figure below). The mass of the mop head is 3.75 kg. Calculate the acceleration of the mop head if the coefficient of kinetic friction between the head and the floor is μk= 0.400. (Express your answer to three significant figures.)
To find the acceleration of the mop head, we need to first determine the net force acting on it parallel to the floor.
First, we find the horizontal (x) and vertical (y) components of the applied force:
Fx = F * cos(Θ) = 36.0 N * cos(50°) = 36.0 N * 0.6428 = 23.141 N
Fy = F * sin(Θ) = 36.0 N * sin(50°) = 36.0 N * 0.7660 = 27.576 N
Next, we find the normal force (N) acting on the mop head, which is simply the net vertical force acting on the mop:
N = mg - Fy = (3.75 kg)(9.81 m/s^2) - 27.576 N = 36.7875 N - 27.576 N = 9.2115 N
Now, we can find the frictional force acting on the mop head:
Friction = μk * N = 0.400 * 9.2115 N = 3.6846 N
Finally, we can determine the net force acting on the mop head parallel to the floor:
Net Force = Fx - Friction = 23.141 N - 3.6846 N = 19.4564 N
Now we can use Newton's second law (F = ma) to calculate the acceleration:
Acceleration = Net Force / Mass = 19.4564 N / 3.75 kg = 5.188 N/kg
Thus, the acceleration of the mop head is 5.19 m/s² (rounded to three significant figures).
To calculate the acceleration of the mop head, we need to determine the net force acting on it.
1. Resolve the applied force into its vertical and horizontal components:
F_horizontal = F * cos(Θ)
F_horizontal = 36.0 N * cos(50.0°)
F_horizontal ≈ 23.05 N
F_vertical = F * sin(Θ)
F_vertical = 36.0 N * sin(50.0°)
F_vertical ≈ 27.56 N
2. Calculate the force of friction acting on the mop head:
F_friction = μk * m * g
where μk is the coefficient of kinetic friction, m is the mass of the mop head, and g is the acceleration due to gravity.
g ≈ 9.8 m/s^2 (acceleration due to gravity)
F_friction = 0.400 * 3.75 kg * 9.8 m/s^2
F_friction ≈ 14.7 N
3. Determine the net force acting on the mop head:
Net force = F_horizontal - F_friction
Net force = 23.05 N - 14.7 N
Net force ≈ 8.35 N
4. Calculate the acceleration of the mop head using Newton's second law:
Net force = mass * acceleration
acceleration = Net force / m
acceleration = 8.35 N / 3.75 kg
acceleration ≈ 2.2267 m/s^2
Therefore, the acceleration of the mop head is approximately 2.23 m/s^2.
To calculate the acceleration of the mop head, we need to consider the forces acting on it.
First, let's find the force of friction. The force of friction can be calculated using the equation:
Friction force = coefficient of friction * normal force
The normal force is the force exerted by the floor on the mop head, which is equal to the mass of the mop head multiplied by the acceleration due to gravity (9.8 m/s^2). Therefore, the normal force can be calculated as:
Normal force = mass * gravity
Next, let's find the net force acting on the mop head. The net force is given by:
Net force = applied force - friction force
Finally, we can calculate the acceleration of the mop head using Newton's second law:
Acceleration = net force / mass
Now, let's calculate the acceleration.
Given:
Applied force (F) = 36.0 N
Angle (Θ) = 50.0°
Mass of the mop head (m) = 3.75 kg
Coefficient of kinetic friction (μk) = 0.400
Acceleration due to gravity (g) = 9.8 m/s^2
1. Calculate the normal force:
Normal force = mass * gravity
Normal force = 3.75 kg * 9.8 m/s^2
Normal force = 36.75 N
2. Calculate the force of friction:
Friction force = coefficient of friction * normal force
Friction force = 0.400 * 36.75 N
Friction force = 14.70 N
3. Calculate the net force:
Net force = applied force - friction force
Net force = 36.0 N - 14.70 N
Net force = 21.30 N
4. Calculate the acceleration:
Acceleration = net force / mass
Acceleration = 21.30 N / 3.75 kg
Acceleration = 5.68 m/s^2
Therefore, the acceleration of the mop head is 5.68 m/s^2.