A ladder of mass M and length L leans against a wall as shown. There is no friction with the wall but there is friction with the ground. What is the maximum distance along the ladder that someone with mass m can climb without the ladder slipping?
To determine the maximum distance along the ladder that someone can climb without the ladder slipping, we need to find the condition where the torque from the weight of the ladder is equal to the torque from the friction force acting at the base of the ladder.
Let's consider the forces acting on the ladder:
1. Weight of the ladder: This force acts downward from the center of mass of the ladder. The magnitude of this force is given by the weight of the ladder, W_ladder = M * g, where M is the mass of the ladder and g is the acceleration due to gravity.
2. Normal force from the ground: This force acts perpendicular to the ground and balances the weight of the ladder.
3. Friction force at the base: This force acts parallel to the ground, opposing any tendency of the ladder to slip.
Now, let's determine the torque from each force acting on the ladder:
1. Torque from the weight of the ladder: The torque is calculated by taking the perpendicular distance between the point of rotation (base of the ladder) and the line of action of the weight force. In this case, the torque is L/2 * W_ladder.
2. Torque from the friction force at the base: The torque is calculated by taking the perpendicular distance between the point of rotation (base of the ladder) and the line of action of the friction force. In this case, the torque is L * F_friction, where F_friction is the magnitude of the friction force.
Since the ladder is in rotational equilibrium (it is not rotating), the torques must balance each other. Therefore, we can equate the torques:
L/2 * W_ladder = L * F_friction
Substituting the values:
L/2 * (M * g) = L * F_friction
Now, we can solve for the friction force:
F_friction = (M * g) / 2
To prevent slipping, the friction force must be greater than or equal to the force required to overcome the gravitational force acting on the climber. In other words, the maximum friction force is equal to the downward force exerted by the climber plus the climber's weight:
F_friction >= (m * g) + (m * g) = 2 * m * g
Substituting the value of F_friction:
(M * g) / 2 >= 2 * m * g
Simplifying the equation by canceling out the gravitational acceleration (g):
M / 2 >= 2 * m
Finally, rearranging the equation to find the maximum distance along the ladder:
M >= 4 * m
Thus, the maximum distance along the ladder that someone with mass m can climb without the ladder slipping is L/4.