A baggage handler drops your 10 kg suitcase onto a conveyor belt running at 2.0 m/s. The materials are such that μ_s = 0.50 and μ_k = 0.30. How far is your suitcase dragged before it is riding smoothly on the belt?

Verified step by step guidance

1. Step 1: Identify the forces acting on the suitcase. The forces include the gravitational force (weight), the normal force, the static friction force, and the kinetic friction force. The gravitational force is given by \( F_g = m \cdot g \), where \( m = 10 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). The normal force is equal in magnitude to the gravitational force since the suitcase is on a horizontal surface.
2. Step 2: Determine the maximum static friction force using \( F_{\text{static}} = \mu_s \cdot F_N \), where \( \mu_s = 0.50 \) and \( F_N = F_g \). If the force required to accelerate the suitcase exceeds this value, the suitcase will transition to kinetic friction.
3. Step 3: Calculate the kinetic friction force using \( F_{\text{kinetic}} = \mu_k \cdot F_N \), where \( \mu_k = 0.30 \). This force will oppose the motion of the suitcase as it is dragged by the conveyor belt.
4. Step 4: Use Newton's second law \( F = m \cdot a \) to find the acceleration of the suitcase while it is under the influence of kinetic friction. The net force acting on the suitcase is \( F_{\text{net}} = F_{\text{kinetic}} \), and \( a = F_{\text{net}} / m \).
5. Step 5: Apply kinematic equations to determine the distance the suitcase is dragged before it matches the speed of the conveyor belt. Use \( v_f^2 = v_i^2 + 2 \cdot a \cdot d \), where \( v_f = 2.0 \, \text{m/s} \), \( v_i = 0 \, \text{m/s} \), and \( a \) is the acceleration calculated in Step 4. Solve for \( d \), the distance dragged.

Key Concepts

Friction

Friction is the force that opposes the relative motion of two surfaces in contact. It is characterized by two coefficients: static friction (μₛ), which prevents motion, and kinetic friction (μₖ), which acts when surfaces are sliding against each other. In this scenario, the suitcase initially experiences static friction until it starts moving with the conveyor belt.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma). This principle is essential for calculating the forces acting on the suitcase as it transitions from rest to moving with the conveyor belt, allowing us to determine how far it is dragged before it matches the belt's speed.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this problem, kinematic equations will be used to calculate the distance the suitcase is dragged before it reaches the same speed as the conveyor belt.