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.
(a) Draw a diagram for the ladder showing all the forces acting on the ladder.
(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.
(a) Diagram:
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Label the forces:
- Fg: Weight of the ladder (750 N) acting downward from the center of mass
- FN: Normal force exerted by the floor perpendicular to the inclined ladder
- FR: Frictional force between the ladder and the floor
- Fw: Force exerted by the wall perpendicular to the ladder
- Fb: Force exerted by the wall parallel to the floor (balanced by FR)
- θ: Angle between the ladder and the wall
(b) Calculation of forces:
- Fg = Weight of the ladder = 750 N (acting downward)
- FN = Normal force exerted by the floor on the ladder = Fg[cos(θ)] (perpendicular to the inclined ladder)
- FR = Frictional force between the ladder and the floor = μ * FN (opposite to the direction of motion)
= 0.40 * FN (opposite to the direction of motion)
- Fw = Force exerted by the wall perpendicular to the ladder (balancing the vertical component of Fg) = Fg[sin(θ)]
- Fb = Force exerted by the wall parallel to the floor (balanced by FR) = FR
(c) In stable equilibrium, the ladder does not rotate. Therefore, the sum of the torques about any point must be zero.
Taking torques about the point where the ladder contacts the floor:
- The torque due to the weight (Fg) = Fg * (length of ladder) * sin(θ) = 750 * 20 * sin(θ)
- The torque due to the force exerted by the wall (Fw) = Fw * (length of ladder) * cos(θ) = Fg[sin(θ)] * (length of ladder) * cos(θ)
Since the ladder is in stable equilibrium, the sum of the torques is zero:
750 * 20 * sin(θ) + Fg[sin(θ)] * 20 * cos(θ) = 0
By solving this equation, the value of θ can be calculated.
(a) The diagram for the ladder showing all the forces acting on it would include the following:
1. The weight of the ladder (750 N): This force acts vertically downward from the center of mass of the ladder.
2. The normal force (N): This force acts vertically upward from the point where the ladder rests on the horizontal surface. It is perpendicular to the surface and counteracts the downward force due to the weight of the ladder.
3. The frictional force (f): This force acts parallel to the horizontal surface and opposes the motion of the ladder. It is responsible for keeping the ladder in equilibrium.
4. The reaction force (R) at the point where the ladder is in contact with the wall: This force acts perpendicular to the wall and counteracts any tendency of the ladder to slide down the wall. It also prevents the ladder from rotating.
5. The force due to the tension in the ladder (T) at the point where it is in contact with the wall: This force acts parallel to the wall and helps keep the ladder in place.
6. The force exerted by the wall on the ladder (W): This force acts perpendicular to the wall and counteracts any component of the weight of the ladder that is directed perpendicular to the wall.
(b) To calculate the numerical values of each force, we need the given information:
Weight of the ladder: 750 N
Coefficient of friction: 0.40
Length of the ladder: 20.0 m
- The weight of the ladder (750 N) acts vertically downward.
- The normal force (N) will be equal to the weight of the ladder, as it counteracts the downward force.
- The frictional force (f) can be calculated using the formula f = coefficient of friction * normal force. In this case, f = 0.40 * N.
- The reaction force (R) at the point where the ladder is in contact with the wall will be equal to the component of the weight of the ladder that is directed perpendicular to the wall. This can be calculated using the formula R = weight of ladder * cos(theta), where theta is the angle between the ladder and the wall.
- The force due to tension (T) can be calculated using the formula T = weight of ladder * sin(theta).
- The force exerted by the wall on the ladder (W) will be equal to the component of the weight of the ladder that is directed perpendicular to the wall. This can be calculated using the formula W = weight of ladder * sin(theta).
(c) To calculate the angle between the ladder and the wall, we can use the equation for equilibrium in the vertical direction:
ΣFy = 0
This implies:
R + W - weight of ladder = 0
Substituting the values, we get:
(weight of ladder * cos(theta)) + (weight of ladder * sin(theta)) - weight of ladder = 0
On simplifying, we can solve this equation to find the value of theta, which represents the angle between the ladder and the wall for stable equilibrium.