A uniform ladder AB of length 6m and mass 40kg rests in a vertical plane with his end B on smooth horizontal ground and the end A against a smooth vertical wall. A mass 20kg is hung from a point on the ladder at a distance 2m from B. The ladder is kept in equilibrium by rope perpendicular to the wall, attach at one edge to the point B and the other end with point C at the foot of the wall. If size BC equal to 3cm, calculate correct to 1 decimal place 1) the reaction of the ground 2) the tension of the rope

To find the reaction of the ground (Force exerted by the ground on the ladder) and the tension of the rope, we can analyze the forces acting on the ladder in equilibrium.

Let's consider the forces acting on the ladder:

1. Weight of the ladder, acting downward from its center of mass (assumed to be at the midpoint of the ladder). The weight of the ladder can be calculated as the product of its mass and gravitational acceleration (9.8 m/s^2).

Weight of the ladder = mass of ladder * gravitational acceleration

Given: mass of the ladder = 40 kg

Weight of the ladder = 40 kg * 9.8 m/s^2

2. Weight of the mass attached to the ladder, acting downward from its point of attachment. Similarly, we can calculate the weight of the mass:

Weight of the mass = mass of the mass * gravitational acceleration

Given: mass of the mass = 20 kg

Weight of the mass = 20 kg * 9.8 m/s^2

Now, let's consider the forces acting on point B of the ladder:

1. The reaction force exerted by the ground vertically upwards.

2. The tension force in the rope, acting horizontally towards the wall.

3. The vertical component of the weight of the ladder (due to gravitational acceleration), acting downward.

4. The vertical component of the weight of the mass, acting downward.

Since the ladder is in equilibrium, the sum of the vertical forces at point B must be zero:

Reaction force of the ground + Vertical component of the weight of the ladder + Vertical component of the weight of the mass = 0

Now, let's express the vertical components of the weights in terms of trigonometric ratios, considering the ladder as a right-angled triangle.

Vertical component of the weight of the ladder = Weight of the ladder * sin(angle between the ladder and the vertical wall)
Vertical component of the weight of the mass = Weight of the mass * sin(angle between the ladder and the vertical wall)

Now, we need to find the angle between the ladder and the vertical wall. Since the ladder is leaning against the wall, this angle can be found using trigonometry:

Angle = inverse tangent (height of the ladder / distance between the ladder and the wall)
Given: height of the ladder = 6 m
distance between the ladder and the wall = 2 m

Angle = inverse tangent (6 m / 2 m)

Now that we have the angle, we can calculate the vertical components of the weights.

Now, considering the horizontal forces at point B:

Tension force in the rope + Horizontal component of the weight of the ladder = 0 (since the ladder is in equilibrium)

Horizontal component of the weight of the ladder = Weight of the ladder * cos(angle between the ladder and the vertical wall)

Now we can solve for the tension in the rope.

To summarize:
1) Calculate the angle between the ladder and the vertical wall using the inverse tangent.
2) Calculate the vertical component of the weight of the ladder and the mass using the calculated angle.
3) Calculate the reaction force of the ground by taking the sum of the vertical weights at point B.
4) Calculate the tension in the rope by taking the negative of the horizontal component of the weight of the ladder.

Plug in the values into the equations to find the answers.