A ladder of mass M and length 6 m rests against a frictionless wall at an angle of 50 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 11M can climb before the ladder starts to slip?
Length L = 6m
Angle with horiontal, θ = 50°
Friction with wall = 0
Coefficient between ladder and floor, μ = 0.5
Mass of ladder = M
Mass of man, m = 11M
Let the man climbs up X m. from the bottom,
Vertical reaction of floor, R = M+11M=12M
(because wall is smooth)
Maximum static friction force to resist movement, F = μR = horizontal reaction on wall
Take moments about the base of the ladder, and for equilibrium, must equal zero:
M*Lcos50+11M*Xcos50-μ*R*Lsin50=0
Solve for X. (between 3 and 4 m.)
To find the maximum distance along the ladder a person of mass 11M can climb before the ladder starts to slip, we need to consider the equilibrium condition for the ladder.
1. Start by drawing a free-body diagram of the ladder.
- There are three forces acting on the ladder: its weight, the normal force from the ground, and the frictional force.
- The weight of the ladder acts downward from its center of mass, located at the center of the ladder.
- The normal force acts upward perpendicular to the ground.
- The frictional force acts parallel to the ground, opposing the motion.
2. Resolve the weight of the ladder into its components.
- The weight of the ladder can be resolved into two components: one parallel to the ground and one perpendicular to the ground.
- The component of the weight perpendicular to the ground does not contribute to the torque, as it acts through the center of mass.
- The component of the weight parallel to the ground provides the torque required to keep the ladder in equilibrium.
3. Set up the torque equation.
- The torque is given by the product of the force and the lever arm.
- The torque from the weight of the ladder is equal to the perpendicular component of the weight multiplied by the lever arm, which is the distance from the wall to the center of mass of the ladder.
- The torque from the frictional force is equal to the magnitude of the frictional force multiplied by the distance from the wall to the point of contact between the ladder and the ground.
4. Equate the torques.
- Set up an equation equating the torques from the weight and the frictional force.
- The torque from the weight of the ladder should be equal to the torque from the frictional force, as the ladder is in equilibrium.
- Solve the equation to find the maximum distance along the ladder.
However, before proceeding with the calculations, it's important to note that the coefficient of static friction given (0.5) is not sufficient to answer the question. We also need the coefficient of friction between the ladder and the wall, denoted by "μw."
Once we have the value for the coefficient of friction between the ladder and the wall, we can proceed with the calculations to find the maximum distance along the ladder.