A uniform ladder of length 20.0m and weight 750 N is propped up against a smooth vertical wall with its lower end on a rough horizontal surface. The coefficient of friction between the ladder and this horizontal surface is 0.40.

(b) Work out and add the numerical values of each force clearly showing your justification in each case.

(c) Hence, calculate a value for the angle between the ladder and the wall if the ladder just remains in stable equilibrium

In order to do part b, the first thing you have to do is consider the horizontal and vertical forces.

For the vertical forces (taking upward as positive) you have the normal reaction at the ground (we'll call this "Ng") which is acting up, and we have the weight of the ladder (750) acting down. These are the only vertical forces.

So we have Ng - 750 = 0 (as it is in equilibrium, the vertical forces must sum to 0)
So Ng = 750 We've now solved one of the three forces you need to label

The second force we can label, the frictional force, actually follows from the value we've just worked out. Recall that friction = coefficient of friction (u/mu) multiplied by the normal reaction (in this case, Ng, or the normal reaction on the floor). mu is given as 0.40 in the question.

So Friction = 0.40 * Ng
= 0.40 * 750
= 300N

Finally, to get the normal reaction at the wall, we consider the horizontal forces. We'll take forces acting the right as positive. We have the normal reaction of the ladder on the wall (we'll call "Nw") acting to the right, and we have the frictional force at the bottom of the ladder acting to the left. From this we get:

Nw - Friction = 0 (in equilibrium, sum to 0)
So, Nw = Friction
But we know the friction is 300N, so Nw = 300N

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For part C, begin by taking moments about the point on the ladder that is touching the wall

Clockwise moments: 750*sin(theta)*10 + 300*cos(theta)*20
Anticlockwise: 750*sin(theta)*20

In equilibrium the clockwise moments = anticlockwise moments

So, 50*sin(theta)*10 + 300*cos(theta)*20 = 750*sin(theta)*20

multiplying out the values we get:

7500sin(theta) + 6000cos(theta) = 1500sin(theta)
7500sin(theta) = 6000cos(theta)

Note that tan(theta) = sin(theta)/cos(theta)

so diving both sides by cos(theta) and then diving through by 7500, we get:

tan(theta) = 6000/7500
solving for theta by taking the tan inverse gives us 38.7

To solve this problem, we need to analyze the forces acting on the ladder. Let's break it down step by step:

Step 1: Identify the forces acting on the ladder.

The forces acting on the ladder are:
1. The weight of the ladder (750 N) acting downward.
2. The normal force from the horizontal surface acting upward.
3. The frictional force between the ladder and the horizontal surface acting horizontally.
4. The normal force from the wall acting perpendicular to the wall.
5. The force exerted by the wall on the ladder acting perpendicular to the wall.
6. The force of static friction between the ladder and the wall acting parallel to the wall.

Step 2: Determine the magnitude and direction of each force.

- Weight of the ladder (750 N): This force acts vertically downward, and its magnitude is given as 750 N.

- Normal force from the horizontal surface: This force opposes the weight of the ladder and prevents it from sinking into the ground. Since the ladder is in stable equilibrium, the force of friction balances the weight, so the normal force from the horizontal surface equals the weight of the ladder, which is 750 N.

- Frictional force between the ladder and the horizontal surface: The frictional force opposes the tendency of the ladder to slip on the horizontal surface. The magnitude of the frictional force can be calculated using the formula: frictional force = coefficient of friction * normal force. Given that the coefficient of friction is 0.40 and the normal force is also 750 N, the frictional force is 0.40 * 750 = 300 N.

- Normal force from the wall: This force is exerted perpendicularly to the wall to prevent the ladder from falling through it. This normal force balances the horizontal component of the force exerted by the ladder on the wall.

- Force exerted by the wall on the ladder: This force acts perpendicular to the wall and opposes the force exerted by the ladder on the wall. Its magnitude is the same as the normal force from the wall.

- Force of static friction between the ladder and the wall: This force opposes the tendency of the ladder to slide down the wall. In equilibrium, this force balances the horizontal component of the force exerted by the ladder on the wall. Its magnitude can be calculated using the formula: force of static friction = coefficient of friction * normal force. Since the ladder is just in stable equilibrium, the force of static friction between the ladder and the wall must be equal to the frictional force between the ladder and the horizontal surface, which is 300 N.

Step 3: Calculate the angle between the ladder and the wall.

To find the angle between the ladder and the wall, we need to consider the forces acting along the length of the ladder. The vertical forces (the weight and the vertical components of the normal forces) cancel out each other in equilibrium. The horizontal forces (the frictional force, the force exerted by the wall, and the horizontal component of the force of static friction) also cancel each other out. This means that the ladder is in equilibrium both vertically and horizontally.

Therefore, the force exerted by the wall must be equal to the horizontal component of the force exerted by the ladder on the wall, which is the force of static friction. So, the magnitude of the force exerted by the wall is 300 N.

Now, we can calculate the angle between the ladder and the wall. This angle can be found using trigonometry. The formula for the tangent of an angle in a right-angled triangle is given by: tangent(angle) = opposite/adjacent.

In this case, the opposite side is the height of the ladder (20 m) and the adjacent side is the distance between the ladder's base and the wall (unknown). Let's call this distance "x."

Using the tangent function, we can write: tangent(angle) = 20/x.

Solving for x, we get: x = 20/tangent(angle).

Since we know that the magnitude of the force exerted by the wall is 300 N, we can substitute this value: 300 = 20/tangent(angle).

Now, we can solve this equation to find the angle between the ladder and the wall. Rearranging the equation, we have: tangent(angle) = 20/300.

Taking the inverse tangent of both sides, we find: angle = arctan(20/300).

Using a calculator, we can evaluate this expression to find the angle.

So, to recap:
- The numerical values of each force are as follows:
- Weight of the ladder: 750 N
- Normal force from the horizontal surface: 750 N
- Frictional force between the ladder and the horizontal surface: 300 N
- Normal force from the wall: 300 N
- Force exerted by the wall on the ladder: 300 N
- Force of static friction between the ladder and the wall: 300 N

- To calculate the angle between the ladder and the wall, we need to use the equation: angle = arctan(20/300). The resulting angle will be in radians, so you might want to convert it to degrees if necessary.