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.
To work out the numerical values of each force, we need to consider the forces acting on the ladder in equilibrium.
First, let's identify the forces acting on the ladder:
1. Weight (W): The weight of the ladder is given as 750 N. This force acts vertically downward from the center of gravity of the ladder.
2. Normal force (N): The normal force is the force exerted by the horizontal surface on the ladder in the upward direction. It acts perpendicular to the surface and balances the weight of the ladder.
3. Friction force (f): The friction force is caused by the coefficient of friction between the ladder and the horizontal surface. It acts parallel to the surface, opposing the ladder's tendency to slide down.
4. Tension force (T): The tension force is the force exerted by the wall on the ladder, keeping it in position. It acts parallel to the ladder and is responsible for preventing it from sliding away from the wall.
Now, let's analyze each force:
1. Weight (W): Its numerical value is given as 750 N.
2. Normal force (N): The normal force is equal in magnitude and opposite in direction to the weight of the ladder (W) since the ladder is in equilibrium. So, N = 750 N.
3. Friction force (f): The friction force can be calculated using the formula f = coefficient of friction × N. Given the coefficient of friction as 0.40 and N as 750 N, we have f = 0.40 × 750 N = 300 N.
4. Tension force (T): The tension force can be calculated using the formula T = W + f. Substituting the given values, T = 750 N + 300 N = 1050 N.
Now, let's move on to calculating the angle between the ladder and the wall for stable equilibrium.
In a ladder in equilibrium, the clockwise and counterclockwise torques must balance. The torque of the weight (W) and friction force (f) acts clockwise, and the torque of the tension force (T) acts counterclockwise.
The torque of a force can be calculated by multiplying the force magnitude by the perpendicular distance from the point of rotation (pivot). In this case, the pivot is the point where the ladder touches the ground, and the perpendicular distance is the length of the ladder divided by 2.
Weight torque: W × (ladder length/2) = 750 N × (20.0 m/2) = 750 N × 10.0 m = 7500 N·m
Friction torque: f × (ladder length/2) = 300 N × (20.0 m/2) = 300 N × 10.0 m = 3000 N·m
Tension torque: T × ladder length = 1050 N × 20.0 m = 21000 N·m
In stable equilibrium, the torques must balance, so the clockwise torques must equal the counterclockwise torque. Therefore, we can calculate the angle between the ladder and the wall using the equation:
T × ladder length = W × (ladder length/2) + f × (ladder length/2)
21000 N·m = 7500 N·m + 3000 N·m
To simplify, we can cancel out the ladder length:
21000 N = 7500 N + 3000 N
Solving for T:
T = 21000 N - 7500 N - 3000 N = 10500 N
Now, we can calculate the angle using the equation:
T = W × (ladder length/2) + f × (ladder length/2)
10500 N = 750 N × (20.0 m/2) + 300 N × (20.0 m/2)
10500 N = 750 N × 10.0 m + 300 N × 10.0 m
10500 N = 7500 N·m + 3000 N·m
10500 N = 10500 N·m
Therefore, the ladder remains in stable equilibrium when the angle between the ladder and the wall is 90 degrees.
To solve this problem, let's first analyze the forces acting on the ladder.
Forces acting on the ladder:
1. Weight of the ladder (Wl): 750 N
2. Normal force exerted by the horizontal surface on the ladder (Nh)
3. Frictional force between the ladder and the horizontal surface (Ff)
4. Force exerted by the wall on the ladder (Fw)
5. Tension in the ladder due to its own weight (T)
(a) Let's calculate the numerical values of each force:
1. Weight of the ladder (Wl): 750 N - This force acts vertically downwards at the center of the ladder.
2. Normal force exerted by the horizontal surface on the ladder (Nh): Since the ladder is in stable equilibrium, the vertical components of forces must balance. Therefore, Nh = Wl = 750 N.
3. Frictional force between the ladder and the horizontal surface (Ff): The frictional force is given by Ff = coefficient of friction * Nh. Substituting the values, Ff = 0.40 * 750 N = 300 N. This force acts horizontally in the opposite direction to the motion of the ladder.
4. Force exerted by the wall on the ladder (Fw): Fw is perpendicular to the ladder as it prevents the ladder from sliding along the wall. This force acts on the ladder at the point of contact with the wall. Since the ladder is in stable equilibrium, the horizontal components of forces must balance. Therefore, Fw = Ff = 300 N.
5. Tension in the ladder due to its own weight (T): The ladder experiences a force due to its weight acting downwards. This force can be resolved into a horizontal component (Th) and a vertical component (Tv). Since the ladder is in stable equilibrium, the horizontal component of T must balance the horizontal forces. Therefore, Th = Fw = 300 N. The vertical component of T must balance the vertical forces. Therefore, Tv = Nh = 750 N.
(b) The numerical values of each force are as follows:
- Weight of the ladder (Wl): 750 N - vertically downwards
- Normal force exerted by the horizontal surface (Nh): 750 N - vertically upwards
- Frictional force between the ladder and the horizontal surface (Ff): 300 N - horizontally opposing the motion
- Force exerted by the wall on the ladder (Fw): 300 N - perpendicular to the wall
(c) Now, let's find the angle between the ladder and the wall when the ladder just remains in stable equilibrium.
Using trigonometry, we can find the angle θ between the ladder and the wall. The vertical component of T (Tv) can be expressed as Tv = T * cos(θ). Rearranging the equation, we have:
T = Tv / cos(θ) = Nh / cos(θ) = 750 N / cos(θ)
To find the value of θ, we equate the torques about the point where the ladder touches the ground to zero. The torques due to Fw, Ff, and Wl are balanced at this point.
Torque due to Fw = Torque due to Ff + Torque due to Wl
Taking torques about the point where the ladder touches the ground,
(Fw * length of the ladder) = (Ff * distance of Ff from the point of contact with the ground) + (Wl * distance of Wl from the point of contact with the ground)
Substituting the values,
(300 N * 20.0 m) = (300 N * ? m) + (750 N * ? m)
Solving this equation will give us the values of the distances. Once we have the distances, we can calculate the angle θ using the equation:
tan(θ) = (distance of Wl) / (distance of Ff)
Substituting the values, we can find the value of θ.