Question

To get up on the roof, a person (mass 92.0 kg) places a 5.60 m aluminum ladder (mass 14.0 kg) against the house on a concrete pad with the base of the ladder 2.00 m from the house. The ladder rests against a plastic rain gutter, which we can assume to be frictionless. The center of mass of the ladder is 2 m from the bottom. The person is standing 3 m from the bottom. What are the magnitudes (in N) of the forces on the ladder at the top and bottom?

Answer 1

Answer:

Down

       F1ₓ = 219.6N

       $F1_{y}$  = 1038.8 N

Top

       F2ₓ = 219.6 N

       $F2_{y}$ = 0  

Explanation:

For this exercise we must make a free body diagram of the ladder, see attached, then use the balance equations on each axis

Transnational Balance

X axis

        F1ₓ -F2ₓ = 0

        F1ₓ = F2ₓ

Y Axis  

         $F1_{y}$ -  $F2_{y}$ - W - W_man = 0           (1)

Rotational balance

The reference system is placed at the bottom of the stairs and we can turn the anti-clockwise direction of rotation as positive

           F2ₓ y - $F2_{y}$ x - W x - W_man x_man = 0

Let us write the data they give, the masses of the ladder (m = 14.0 kg), the mass of man (m_man = 92 kg), the center of mass of the ladder that is 2m from the bottom (the height) and the position of the man which is 3 m high

Let's look with trigonometry for distances

The angle of the stairs is

           cos θ = x / L

           θ = cos⁻¹ x / L

           θ = cos⁻¹ 2 / 5.6

           θ = 69⁰

Height y

          tan 69 = y / x

          y = x tan 69

          y = 2 tan 69

          y = 5.21 m

Distance x

          tan 69 = 2 / x

          x = 2 / tan 69

          x = 0.7677 m

The distance x_man

          x_man = 3 / tan 69

          x_man = 1,152 m

They indicate that between the scalars and the support there is no friction so the vertical force at the top is zero

          $F2_{y}$ = 0

Let's replace in the translational equilibrium equation

         F2ₓ y - $F2_{y}$ x - W x - W_man x_man = 0

         F2ₓ 5.21 -0 - 14.0 9.8 0.7677 - 92.0 9.8 1,152 = 0

         F2ₓ = 1143.97 / 5.21

         F2ₓ = 219.6 N

 

We use equation 1

         $F1_{y}$ + 0 - W - W_man = 0

        $F1_{y}$ = W + W_man

        $F1_{y}$  = (m + m_man) g

         $F1_{y}$  = (14 +92) 9.8

         $F1_{y}$  = 1038.8 N

We can write the force on each part of the ladder

Down

       F1ₓ = 219.6N

       $F1_{y}$  = 1038.8 N

Top

       F2ₓ = 219.6 N

       $F2_{y}$ = 0