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. Step 1: Begin by analyzing the forces acting on the toolbox. Draw a free-body diagram of the toolbox. The forces include: (1) the gravitational force acting downward, \( F_g = m g \), (2) the normal force acting upward, \( F_N \), and (3) the frictional force acting horizontally, which prevents the toolbox from sliding. The frictional force is determined by the static friction coefficient \( \mu_s \).
2. Step 2: Use Newton's second law to analyze the horizontal forces. The maximum static friction force \( F_{f, \text{max}} \) is given by \( F_{f, \text{max}} = \mu_s F_N \). Since the toolbox is not sliding, the normal force \( F_N \) equals the gravitational force \( F_g \), so \( F_{f, \text{max}} = \mu_s m g \).
3. Step 3: Relate the frictional force to the acceleration of the truck. The toolbox will remain stationary relative to the truck as long as the acceleration \( a \) does not exceed the maximum static friction force divided by the mass of the toolbox. Using \( F = ma \), the maximum acceleration is \( a_{\text{max}} = \frac{F_{f, \text{max}}}{m} = \mu_s g \).
4. Step 4: Calculate the shortest time required for the truck to accelerate uniformly to 30.0 m/s without causing the toolbox to slide. Use the kinematic equation \( v = a t \), where \( v \) is the final velocity, \( a \) is the acceleration, and \( t \) is the time. Rearrange to solve for \( t \): \( t = \frac{v}{a} \). Substitute \( a_{\text{max}} = \mu_s g \) into the equation.
5. Step 5: Substitute the known values into the equations. Use \( \mu_s = 0.650 \), \( g = 9.8 \, \text{m/s}^2 \), and \( v = 30.0 \, \text{m/s} \) to calculate the time \( t \). This will give the shortest time the truck can accelerate uniformly to 30.0 m/s without causing the toolbox to slide.

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, which prevents motion until a certain threshold is reached, and kinetic friction, which acts when surfaces are sliding past each other. In this scenario, the static friction coefficient (0.650) is crucial for determining the maximum acceleration the truck can have without the toolbox sliding.

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 toolbox and determining the maximum acceleration the truck can achieve without exceeding the static friction limit.

Free-Body Diagram

A free-body diagram is a graphical representation that shows all the forces acting on an object. In this case, it would illustrate the gravitational force acting downward on the toolbox, the normal force acting upward, and the frictional force opposing the motion. This diagram is vital for visualizing the forces involved and applying Newton's laws to solve for the truck's acceleration.