A house painter stands 3 m above the ground on a 5.3-m-long ladder that leans against the wall at a point 4.7 m above the ground. The painter weighs 638 N and the ladder weighs 143 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.
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.
There are three forces acting on the ladder: the weight of the painter (P), the weight of the ladder (W), and the force of friction (F) at the bottom of the ladder.
First, let's calculate the weight of the painter and the ladder:
Weight of the painter (P) = mass of the painter x acceleration due to gravity
= 638 N (given)
Weight of the ladder (W) = mass of the ladder x acceleration due to gravity
= 143 N (given)
Next, let's consider the equilibrium of forces along the ladder.
The weight of the painter (P) can be resolved into two components: one parallel to the ladder and one perpendicular to the ladder. The component parallel to the ladder balances the force of friction, while the component perpendicular to the ladder balances the tension in the ladder.
The perpendicular component of the weight of the painter = P x (h/l)
= 638 N x (3 m / 5.3 m)
= 361.13 N
The perpendicular component of the weight of the ladder = W x (h/l)
= 143 N x (3 m / 5.3 m)
= 80.47 N
Now, we can calculate the force of friction (F):
Force of friction (F) = Perpendicular component of the weight of the painter + Perpendicular component of the weight of the ladder
= 361.13 N + 80.47 N
= 441.6 N
Therefore, the force of friction that the driveway exerts on the bottom of the ladder is 441.6 N.