A pickup truck is carrying a toolbox, but the rear gate of the truck is missing. The toolbox will slide out if it is set moving. The coefficients of kinetic friction and static friction between the box and the level bed of the truck are 0.355 and 0.650, respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to 30.0 m/s without causing the box to slide? Draw a free-body diagram of the toolbox.

Verified step by step guidance

1. First, identify the forces acting on the toolbox. The forces include the gravitational force (weight) acting downward, the normal force acting upward, and the frictional force acting horizontally opposite to the direction of acceleration.
2. Draw a free-body diagram of the toolbox. Represent the gravitational force \( F_g = m \cdot g \) acting downward, the normal force \( F_n \) acting upward, and the static frictional force \( F_s \) acting horizontally. The static frictional force is what prevents the toolbox from sliding.
3. Calculate the maximum static frictional force using the formula \( F_s = \mu_s \cdot F_n \), where \( \mu_s = 0.650 \) is the coefficient of static friction and \( F_n = m \cdot g \) is the normal force. This force must be greater than or equal to the force required to accelerate the toolbox to prevent it from sliding.
4. Determine the maximum acceleration \( a_{max} \) that the truck can have without the toolbox sliding. Use the equation \( F_s = m \cdot a_{max} \) and solve for \( a_{max} \).
5. Finally, use the kinematic equation \( v = u + a \cdot t \) to find the shortest time \( t \) for the truck to accelerate to 30.0 m/s, where \( v = 30.0 \text{ m/s} \), \( u = 0 \text{ m/s} \), and \( a = a_{max} \). Solve for \( t \).

Key Concepts

Static and Kinetic Friction

Static friction is the force that keeps an object at rest and prevents it from moving, while kinetic friction acts on an object in motion. The static friction coefficient (0.650) is higher than the kinetic friction coefficient (0.355), indicating that more force is needed to start moving the toolbox than to keep it moving. Understanding these forces is crucial to determine the maximum acceleration before the toolbox starts sliding.

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle helps calculate the maximum acceleration the truck can have without the toolbox sliding, by equating the force of static friction to the product of the toolbox's mass and the truck's acceleration.

Uniform Acceleration

Uniform acceleration refers to a constant acceleration over time. In this scenario, the truck accelerates uniformly to reach a speed of 30.0 m/s. By using the equations of motion, specifically the formula v = u + at, where v is the final velocity, u is the initial velocity, a is acceleration, and t is time, we can determine the shortest time required for this acceleration without the toolbox sliding.