A 77 kg window cleaner uses a 13 kg ladder that is 7.1 m long. He places one end on the ground 3.8 m from a wall, rests the upper end against a cracked window, and climbs the ladder. He is 3.9 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 magnitude of the force on the window from the ladder, we need to analyze the forces acting on the ladder. There are two forces: the weight of the ladder and the weight of the window cleaner.
First, let's calculate the weight of the ladder. The weight of an object is calculated by multiplying its mass by the acceleration due to gravity (9.8 m/s²):
Weight of ladder = mass of ladder × gravitational acceleration
= 13 kg × 9.8 m/s²
= 127.4 N
Next, let's calculate the weight of the window cleaner:
Weight of window cleaner = mass of window cleaner × gravitational acceleration
= 77 kg × 9.8 m/s²
= 754.6 N
Now, let's consider the forces acting on the ladder. There is a vertical component due to the weight of the ladder plus the weight of the window cleaner, and there is a horizontal component due to the reaction force from the wall.
The vertical component of the force on the window from the ladder is equal in magnitude and opposite in direction to the vertical component of the weight of the ladder and the window cleaner:
Vertical force on the window = (weight of ladder + weight of window cleaner) × cos(angle)
= (127.4 N + 754.6 N) × cos(angle)
Given that the vertical distance from the ground to the point of application of the force is 3.9 m and the horizontal distance is 3.8 m, we can use trigonometry to find the angle:
tan(angle) = vertical distance / horizontal distance
angle = arctan(vertical distance / horizontal distance)
angle = arctan(3.9 m / 3.8 m)
(a) The magnitude of the force on the window from the ladder is (127.4 N + 754.6 N) × cos(angle).
(b) The magnitude of the force on the ladder from the ground is (127.4 N + 754.6 N) × sin(angle).
(c) The angle relative to the horizontal of that force on the ladder is arctan(3.9 m / 3.8 m).
To find the magnitude of the force on the window from the ladder, we can use the principle of equilibrium.
Step 1: Calculate the weight of the window cleaner and the ladder.
The weight of the window cleaner is given as 77 kg. The weight of the ladder is given as 13 kg.
Weight = mass * gravitational acceleration
Weight of the window cleaner = 77 kg * 9.8 m/s^2 = 754.6 N
Weight of the ladder = 13 kg * 9.8 m/s^2 = 127.4 N
Step 2: Calculate the torque exerted on the ladder by the window cleaner.
Torque = force * perpendicular distance from the point of rotation
In this case, the point of rotation is the base of the ladder on the ground.
The perpendicular distance from the point of rotation to the weight of the window cleaner is 3.8 m.
Torque = Weight * perpendicular distance
Torque = 754.6 N * 3.8 m = 2865.48 N*m
Step 3: Calculate the torque exerted on the ladder by the ground.
When the ladder is on the verge of breaking, the net torque on the ladder will be zero.
The torque exerted by the ground will be opposite in direction to the torque exerted by the window cleaner.
Torque exerted by the ground = - Torque exerted by the window cleaner
Torque exerted by the ground = -2865.48 N*m
Step 4: Calculate the force exerted by the ground on the ladder.
We can use the formula torque = force * perpendicular distance from the point of rotation.
The perpendicular distance from the point of rotation to the force exerted by the ground is 7.1 m.
Torque exerted by the ground = force * 7.1 m
-2865.48 N*m = force * 7.1 m
force = -2865.48 N*m / 7.1 m ≈ -403.6 N
(a) The magnitude of the force on the window from the ladder is approximately 403.6 N.
(b) The magnitude of the force on the ladder from the ground is 403.6 N.
(c) The angle (relative to the horizontal) of that force on the ladder can be found using trigonometry:
sin(angle) = perpendicular distance / hypotenuse
sin(angle) = 3.8 m / 7.1 m
angle = sin^(-1)(3.8 m / 7.1 m) ≈ 30.7 degrees.
Therefore, the angle of the force on the ladder from the ground is approximately 30.7 degrees.