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
at top
force to left from wall = Fxt
vertical force from wall = 0
at bottom
Force up from floor = Fzb
max force to right from floor =Fxb
= .4 Fzb
Gravity force down at 10 meters from base = 750 N
not accelerating so
Fzb = 750
Fxb = .4(750)
so
Fxt also = .4(750) but to left
angle to floor = T
take moments about center of ladder
Fzb (10 cos T) = Fxb (10 sin t) + Fxt(10 sin T)
750 cos T = 2 (.4*750) sin T
tan T = 1/.8
To calculate the forces acting on the ladder, we need to consider the different forces involved: the weight of the ladder, the normal force from the ground, the frictional force, and the force exerted by the wall.
(b) Let's break down the forces and calculate their values:
1. Weight of the ladder (W): Given as 750 N.
2. Normal force from the ground (N): This force acts perpendicular to the ground's surface and supports the weight of the ladder. Since the ladder is in equilibrium (not moving up or down), it is equal in magnitude to the weight of the ladder, so N = 750 N.
3. Frictional force (F_friction): The coefficient of friction between the ladder and the horizontal surface is given as 0.40. The frictional force can be calculated using the equation F_friction = coefficient of friction × N. Substituting the values, we get F_friction = 0.40 × 750 N = 300 N.
4. Force exerted by the wall (F_wall): This force acts perpendicular to the wall and prevents the ladder from sliding down. By Newton's third law, it is equal in magnitude but opposite in direction to the horizontal component of the force exerted by the ladder onto the wall.
To calculate the angle between the ladder and the wall, we need to find the horizontal component of the force exerted by the ladder onto the wall. Assuming the ladder is in stable equilibrium, this horizontal component can be balanced by the frictional force.
The ladder can be considered as a simple beam in equilibrium. Using this, we can write the equation for equilibrium as follows:
Sum of torques = 0
The torque due to the ladder's weight (W) about the bottom end can be calculated as (W × length) × sin(theta), where theta is the angle between the ladder and the wall.
Similarly, the torque due to the frictional force (F_friction) about the bottom end can be calculated as (F_friction × effective distance) × sin(theta), where the effective distance is the distance from the bottom end to the point where the frictional force acts.
Since the ladder is just in stable equilibrium, the torques due to the weight and the frictional force must be equal. Therefore, we can equate these torques and solve for theta.
(c) Substituting the known values, the equation becomes:
(W × length) × sin(theta) = (F_friction × effective distance) × sin(theta)
(750 N × 20.0 m) × sin(theta) = (300 N × effective distance) × sin(theta)
By canceling out sin(theta), we have:
(750 N × 20.0 m) = (300 N × effective distance)
From this equation, we can solve for the effective distance.
Finally, by using the effective distance and the given ladder length, we can calculate the angle between the ladder and the wall by taking the inverse sine of (effective distance / ladder length).