A window washer pushes his scrub brush up a vertical window at constant speed by applying a force F as shown in the figure below, making angle of 42 degree with horizontal line. The brush has a mass of 1.6 kg and the coefficient of kinetic friction is 0.23 with the window. Find the magnitude
of the force F?
To find the magnitude of the force F, we need to analyze the forces acting in the vertical direction (up and down).
There are three forces acting on the scrub brush in the vertical direction:
1. The force of gravity (mg), where m is the mass of the brush (1.6 kg) and g is the acceleration due to gravity (9.8 m/s^2). This force acts downward.
2. The normal force (N) exerted by the window on the brush. This force acts perpendicular to the window and cancels out the vertical component of the force of gravity.
3. The force of kinetic friction (f_k) between the brush and the window. This force opposes the motion and acts in the opposite direction of F.
Since the brush is moving at constant speed, the net force in the vertical direction is zero. This means that the force F must balance out the combined vertical forces of gravity and friction.
Now, let's break down the forces and set up the equation:
1. The vertical component of gravity: mg * sin(42°)
2. The force of kinetic friction: f_k = μ * N, where μ is the coefficient of kinetic friction (0.23)
Since the net vertical force is zero, we have the following equation:
F - mg * sin(42°) - f_k = 0
Now, substitute the values:
F - (1.6 kg) * (9.8 m/s^2) * sin(42°) - (0.23) * N = 0
To solve for F, we need to find the value of the normal force N. N is equal to the vertical component of the force of gravity (mg * cos(42°)):
N = (1.6 kg) * (9.8 m/s^2) * cos(42°)
Substitute the value of N back into the equation and solve for F:
F - (1.6 kg) * (9.8 m/s^2) * sin(42°) - (0.23) * (1.6 kg) * (9.8 m/s^2) * cos(42°) = 0
Now, calculate the value of F using the given values and solve the equation.