A ladder of mass M and length 2 m rests against a frictionless wall at an angle of 47 degrees to the horizontal. The coefficient of static friction between the ladder and the floor is 0.5. What is the maximum distance along the ladder a person of mass 13M can climb before the ladder starts to slip?
i sitll have no idea how to do this. please help!
Draw the free-body diagram.
Pivot is on the floor.
Normal force of floor is pointing up(Nf).
Friction is in positive direction on the horizontal(f).
Normal of the wall is pointing in negative direction(Nw).
+ for counterclockwise, - for clockwise motion
remember that the θ in rFsinθ is the angle between r and F so draw the ladder cutting into the wall so you can draw the angle between the ladder and Nw.
ΣFy=0; Nf-13Mg-Mg=0 ∴ Nf=14Mg
ΣFx=0; -Nw+f=0 ∴ Nw=f
Nw=f=μNf=[μ14Mg] (you'll need this later)
Στ=0; Nf(0)+[Nw]sin133°+(d-(2/2)13Mgsin137°-(2/2)Mgsin137°=0
replace Nw and solve for d.
oops my bad, the weight of the person is in clockwise direction so it's negative
Στ=0; Nf(0)+[Nw]sin133°-(d-(2/2))13Mgsin137°-(2/2)Mgsin137°=0
To find the maximum distance along the ladder a person can climb before it starts to slip, we need to determine the point at which the ladder is on the verge of sliding. This occurs when the static friction force between the ladder and the floor reaches its maximum value.
Let's break down the problem step by step:
1. Calculate the Normal Force:
The ladder exerts a force on the ground at its base, which creates a normal force in the opposite direction. In this case, the normal force is equal to the weight of the ladder plus the weight of the person. Therefore, the normal force (N) is given by:
N = (M + 13M) * g, where g is the acceleration due to gravity.
2. Determine the Maximum Static Friction Force:
The maximum static friction force (F_max) is found by multiplying the coefficient of static friction (μ_s) by the normal force:
F_max = μ_s * N
3. Calculate the Torque:
The torque created by the weight of the ladder and the person must be balanced by the torque created by the frictional force. The torque exerted by the weight is given by:
τ_weight = 2m * (M + 13M) * g * sin(47°)
4. Find the Maximum Distance:
The maximum distance (d_max) along the ladder a person can climb before it starts to slip occurs when the torque exerted by the frictional force is equal to the torque exerted by the weight. Therefore, we can set the two torques equal to each other:
F_max * d_max = τ_weight
Solving for d_max, we have:
d_max = τ_weight / F_max
Substituting the torque and friction force equations into the above expression, we get:
d_max = (2m * (M + 13M) * g * sin(47°)) / (μ_s * (M + 13M) * g)
Simplifying the expression, the masses and acceleration due to gravity cancel out, and we are left with:
d_max = (2m * sin(47°)) / μ_s
Now, plug in the given values:
d_max = (2 * 2 * sin(47°)) / 0.5
Calculate the result, and we find:
d_max = 6.28 meters
Therefore, the maximum distance along the ladder a person of mass 13M can climb before the ladder starts to slip is approximately 6.28 meters.