Question

A ladder of mass \(M\) and length \(L=4.00 \mathrm{~m}\) is on a level floor leaning against a vertical wall. The coefficient of static friction between the ladder and the floor is \(\mu_{\mathrm{s}}=\) 0.600 , while the friction between the ladder and the wall is negligible. The ladder is at an angle of \(\theta=50.0^{\circ}\) above the horizontal. A man of mass \(3 M\) starts to climb the ladder. To what distance up the ladder can the man climb before the ladder starts to slip on the floor?

Short answer

Answer: The man can climb up to approximately 1.33 meters up the ladder before it starts to slip.

Step-by-step solution

Calculate the weight of ladder and man

First, calculate the gravitational force acting on the ladder and man, i.e., the weight of ladder and man. Weight of ladder (W_L) = Mass of ladder × Gravity W_L = M × g Weight of man (W_M) = Mass of man × Gravity W_M = 3M × g Where M is the mass of ladder, g is the acceleration due to gravity, and g ≈ 9.81 m/s².

Find the horizontal and vertical components of forces

Next, determine the horizontal (H) and vertical (V) components of the forces acting on the ladder. The horizontal component, H, is the frictional force between the ladder and the floor: H = F_friction = μ_s × N = μ_s × (W_L + W_M) The vertical component, V, is the sum of the normal force acting between the ladder and the wall and the wall's normal force acting on the ladder: V = N_wall + W_L + W_M Here, μ_s is the coefficient of static friction and N is the normal force.

Apply the torque equation to find the slipping distance

To find the slipping distance, apply the torque (τ) about the contact point between the ladder and the floor. Use the x-component of the torques acting on the ladder to set up the equation for the maximum slipping distance: τ_max = W_L × L × sin(θ) = W_M × distance × sin(θ) Here, θ is the angle between the ladder and the floor. Rearranging to get the distance before slipping: distance = (W_L × L) / W_M Substitute the values for W_L and W_M: distance = (M × g × L) / (3M × g)

Calculate the distance before slipping

Finally, plug in the given values of M, L, and g to calculate the distance: distance = (4m) / 3 distance ≈ 1.33 m The man can climb up to 1.33 meters up the ladder before it starts to slip on the floor.

Key concepts

Static Friction

Static friction is a force that resists the initial movement of two objects in contact with each other when one attempts to move over the other. It's what keeps the ladder in place on the floor and is a crucial concept in physics problem-solving. The coefficient of static friction, represented by \(\mu_s\), is a dimensionless number describing the frictional force between two static objects. The higher the coefficient, the greater the frictional resistance.

In this exercise, when the man starts to climb the ladder, the static friction between the ladder's base and the floor opposes the lateral force due to the man's weight. Knowing the coefficient of static friction allows us to calculate the maximum horizontal force before slipping, by multiplying it with the normal force — the force perpendicular to the contact surface. In equilibrium, the maximum static friction and gravitational force components parallel to the floor must balance for no motion to occur. Once exceeded, the ladder would slip, leading to the scenario the problem aims to predict.

Torque and Equilibrium

Torque is the rotational equivalent of force that causes objects to turn or rotate around an axis. For an object to be in rotational equilibrium, the total torque acting on it must be zero. In the ladder problem, we're considering the torques about the contact point on the floor. Torque is calculated as the product of the force and the perpendicular distance from the rotation axis to the direction of the force — a concept exemplified by \(\tau = W \times d \times \sin(\theta)\), where \(\tau\) denotes the torque, \(W\) is the force' weight, \(d\) is the distance along the ladder, and \(\theta\) is the angle of the ladder with the ground.

The equilibrium condition indicates that the torque due to the ladder's weight and the torque due to the man's weight have to balance each other. The torque analysis allows us to solve for the maximum height the man can climb before the torque due to his weight causes a torque imbalance, leading to the ladder slipping.

Forces in Physics

Forces are interactions that, if unopposed, lead to a change in motion of an object. In our ladder scenario, the concept of forces in physics enables a comprehensive understanding of the problem by considering a few types of forces: the force of gravity and the reaction forces from the wall and floor. The gravitational force acts downward, while reaction forces include the normal force from the ground and friction. These forces must be in equilibrium for the ladder to remain stationary.

Understanding forces is integral to physics problem-solving as it lets us dissect complex scenarios into calculable components. The force of gravity is always present and acts at the center of mass, which affects the system's stability. Properly resolving these forces into perpendicular components is key to solving equilibrium problems. By calculating these forces and their points of application, we can apply Newton's laws of motion and determine the conditions for stasis or movement.