A 75 kg window cleaner uses a 10 kg ladder that is 7.2 m long. He places one end on the ground 4.5 m from a wall, rests the upper end against a cracked window, and climbs the ladder. He is 5.2 m up along the ladder when the window breaks. Neglect friction between the ladder and window and assume that the base of the ladder does not slip. When the window is on the verge of breaking, what are (a) the magnitude of the force on the window from the ladder, (b) the magnitude of the force on the ladder from the ground, and (c) the angle (relative to the horizontal) of that force on the ladder?
To find the answers to the given questions, we need to analyze the forces acting on the ladder and the window cleaner. Let's start by breaking down the problem:
(a) The magnitude of the force on the window from the ladder: This force can be determined by considering the forces acting on the ladder in equilibrium. When the ladder is about to break the window, the ladder is not slipping, so the net force on it must be zero. The force on the window from the ladder is equal in magnitude and opposite in direction to the force on the ladder from the window.
(b) The magnitude of the force on the ladder from the ground: This force can be determined by considering the vertical equilibrium of forces acting on the ladder. Since the ladder is not slipping, the vertical forces must balance.
(c) The angle (relative to the horizontal) of the force on the ladder: We will need the inverse tangent function to find this angle.
Now, let's calculate the values step by step:
(a) Finding the magnitude of the force on the window from the ladder:
Since we know the mass of the window cleaner (75 kg) and the length of the ladder (7.2 m), we can calculate the center of mass of the ladder using the formula:
Center of Mass = (Mass of ladder * Length of ladder) / (Mass of ladder + Mass of window cleaner)
Center of Mass = (10 kg * 7.2 m) / (10 kg + 75 kg)
Center of Mass = 0.86 m
The force on the window from the ladder is equal to the weight of the window cleaner acting on the center of mass of the ladder:
Force on window from ladder = Mass of window cleaner * gravitational acceleration
Force on window from ladder = 75 kg * 9.8 m/s^2
Force on window from ladder = 735 N
(b) Finding the magnitude of the force on the ladder from the ground:
Since the ladder is not slipping, the vertical forces must balance.
The vertical forces acting on the ladder are the weight of the ladder, the weight of the window cleaner, and the force from the ground.
Weight of ladder = Mass of ladder * gravitational acceleration
Weight of ladder = 10 kg * 9.8 m/s^2
Weight of ladder = 98 N
Weight of window cleaner = Mass of window cleaner * gravitational acceleration
Weight of window cleaner = 75 kg * 9.8 m/s^2
Weight of window cleaner = 735 N
Let's assume the force from the ground is F. The vertical equilibrium equation becomes:
Force from the ground = Weight of ladder + Weight of window cleaner
F = 98 N + 735 N
F = 833 N
(c) Finding the angle of the force on the ladder:
To find the angle, we will use the inverse tangent function:
Angle = arctan(height/length)
Angle = arctan(5.2 m / 4.5 m)
Angle = arctan(1.16)
Angle ≈ 49.5°
Therefore, the answers are:
(a) The magnitude of the force on the window from the ladder is 735 N.
(b) The magnitude of the force on the ladder from the ground is 833 N.
(c) The angle (relative to the horizontal) of that force on the ladder is approximately 49.5°.