Window Washer Force Problem – Friction and Inclined Forces

Q19. A window washer pushes his scrub brush up a vertical window at constant speed by applying a force \( \vec{F} \) as shown in the figure. The brush weighs 12.0 N and the coefficient of kinetic friction is 0.150. The rod is very light, so its mass is ignored. What is the magnitude of the force \( \vec{F} \)?

Background

Topic: Forces and Friction on an Inclined Plane

This question tests your understanding of how to resolve forces when an object is pushed at an angle, including the effects of friction and gravity. The brush moves at constant speed, so the net force along the direction of motion is zero.

Key Terms and Formulas

• Weight (W): The force due to gravity, \( W = mg \), here given as 12.0 N.

• Kinetic Friction (f_k): \( f_k = \mu_k N \), where \( \mu_k \) is the coefficient of kinetic friction and \( N \) is the normal force.

• Force Components: The applied force \( \vec{F} \) can be broken into vertical and horizontal components using the angle \( \theta \).

• Constant Speed: Implies net force along the direction of motion is zero (Newton's First Law).

Step-by-Step Guidance

1. Draw a free-body diagram for the brush, showing the weight (downward), friction (opposing motion), normal force (perpendicular to the window), and the applied force \( \vec{F} \) at 53.1° above the horizontal.

2. Resolve the applied force \( \vec{F} \) into components: \( F_x = F \cos(53.1^) \) (horizontal, into the window) \( F_y = F \sin(53.1^) \) (vertical, upward along the window)

3. Write the equilibrium equations for vertical and horizontal directions. Since the brush moves at constant speed, the sum of forces in each direction must be zero.

4. Set up the vertical force balance: \( F_y = W + f_k \) The vertical component of \( \vec{F} \) must overcome both the weight and the friction force.

5. Express the friction force in terms of the normal force: \( f_k = \mu_k N \) The normal force is determined by the horizontal component of \( \vec{F} \): \( N = F_x = F \cos(53.1^) \)

Try solving on your own before revealing the answer!