A house painter stands 3.0 m above the ground on a
5.0-m-long ladder that leans against the wall at a point 4.7 m above the
ground. The painter
weighs 680 N and
the ladder weighs
120 N. Assuming no
friction between the
house and the upper
end of the ladder,
find the force of
friction that the
driveway exerts on
the bottom of the
ladder. (
✦
✦
2.4 m
1.2 m
FA
F
3.0 m
0.50 m
1200 N
1.80 m
CG
1.10 m
q
d
FL FR
0.8600 m
0.3333 m
To find the force of friction that the driveway exerts on the bottom of the ladder, we need to consider the forces acting on the ladder.
First, let's label the relevant distances and forces:
- Distance from the ground to the bottom of the ladder: d = 3.0 m
- Distance from the ground to the top of the ladder: 4.7 m
- Length of the ladder: 5.0 m
- Weight of the painter: 680 N
- Weight of the ladder: 120 N
- Force of friction on the bottom of the ladder: Ff
Now, let's analyze the forces acting on the ladder:
1. Weight of the painter: The weight of the painter acts downward at the center of gravity (CG) of the system. The magnitude of this force is 680 N.
2. Weight of the ladder: The weight of the ladder acts downward at the center of gravity (CG) of the ladder. The magnitude of this force is 120 N.
3. Normal force: The normal force acts upward at the bottom of the ladder, supporting the weight of the ladder and the painter. The magnitude of this force is equal to the sum of the weight of the painter and the ladder, which is 800 N (680 N + 120 N).
4. Friction force: The force of friction acts parallel to the surface of the driveway, opposing the motion of the ladder. This is the force we need to find.
To find the force of friction, we need to use the concept of torque. The torque equation is given by:
Torque = Force x Distance
Since the ladder is in equilibrium (not rotating), the sum of the torques acting on the ladder must be zero.
Now, let's calculate the torques:
1. Torque due to the weight of the painter (680 N): The torque is calculated by multiplying the weight of the painter by the perpendicular distance from the CG of the system to the bottom of the ladder. The perpendicular distance is 1.2 m, so the torque is 680 N x 1.2 m = 816 Nm.
2. Torque due to the weight of the ladder (120 N): The torque is calculated by multiplying the weight of the ladder by the perpendicular distance from the CG of the ladder to the bottom of the ladder. The perpendicular distance is 2.4 m, so the torque is 120 N x 2.4 m = 288 Nm.
3. Torque due to the force of friction (Ff): The torque is calculated by multiplying the force of friction by the perpendicular distance from the bottom of the ladder to the point of contact with the ground. The perpendicular distance is d, which is 3.0 m, so the torque is Ff x 3.0 m.
Since the ladder is in equilibrium (not rotating), the sum of the torques must be zero:
816 Nm + 288 Nm + Ff x 3.0 m = 0
Simplifying the equation:
1104 Nm + Ff x 3.0 m = 0
Rearranging the equation to solve for Ff:
Ff x 3.0 m = -1104 Nm
Ff = -1104 Nm / 3.0 m
Ff = - 368 N
The negative sign indicates that the force of friction is acting in the opposite direction of the assumed positive direction. However, since the question asks for the magnitude of the force of friction, we can ignore the negative sign. Therefore, the force of friction exerted by the driveway on the bottom of the ladder is approximately 368 N.