Question

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.

Short answer

The shortest time is approximately 4.70 seconds.

Step-by-step solution

Identify the Forces Acting on the Toolbox

To solve this problem, first consider the forces acting on the toolbox. The forces include the force of gravity pulling it downwards, the normal force from the truck bed acting upwards, and the static frictional force acting horizontally to prevent it from sliding.- Gravitational Force = \(mg\)- Normal Force = \(N\)- Maximum Static Frictional Force = \(f_s = \mu_s N\)

Establish the Conditions for Static Friction

The toolbox will not slide if the static friction force can resist the horizontal force due to acceleration. Since the maximum static frictional force is \(f_s = \mu_s mg\), this force must be equal or greater than the force required to move the toolbox at the truck's acceleration:\(f_s \geq ma\)So, \(\mu_s mg \geq ma\)

Calculate Maximum Possible Acceleration

To find the maximum acceleration \(a\) without the toolbox sliding, rearrange the inequality from Step 2:\[\mu_s mg = ma\]Simplifying gives:\[a = \mu_s g\]Plugging in \(\mu_s = 0.650\) and \(g = 9.81 \, \text{m/s}^2\), we can calculate:\[a = 0.650 \times 9.81 = 6.3765 \, \text{m/s}^2\]

Calculate the Shortest Time to Reach Desired Speed

Using the formula for constant acceleration, we know:\[v = at\]where \(v = 30.0 \, \text{m/s}\), and solving for \(t\) gives:\[t = \frac{v}{a} = \frac{30.0}{6.3765} \approx 4.70 \, \text{seconds}\]

Construct the Free-Body Diagram

Draw a box representing the toolbox. Indicate the force of gravity acting downwards, the normal force acting upwards, the maximum static frictional force acting horizontally to the left, resisting the movement to the right.

Key concepts

Static Friction

Static friction acts between two surfaces that are not moving relative to each other. It is the force that keeps the toolbox from sliding off the truck when the truck accelerates. Static friction must be overcome for an object to start moving.
The force of static friction depends on:

• The nature of the surfaces (roughness, etc.)
• The normal force (weight of the object)

It can be calculated using the formula:

• Maximum static frictional force: \( f_s = \mu_s N \)
• \( \mu_s \): coefficient of static friction
• \( N \): normal (support) force

The concept is crucial because it determines whether the toolbox will slide off or remain on the truck as it accelerates. If the force from acceleration exceeds static friction, the toolbox will slide.

Free-Body Diagram

A free-body diagram is a simple graphical illustration used to visualize the forces acting on an object.
In the context of the toolbox:

• Draw a box to represent the toolbox
• Show the gravitational force (\( mg \)) acting downwards
• Show the normal force (\( N \)) acting upwards
• Indicate the static frictional force horizontally opposing potential sliding

This diagram helps in analyzing the problem by visualizing all forces involved. It simplifies the complex interactions into a clear picture, making it easier to apply Newton's laws of motion.
By balancing the forces, we can determine if and when the box will start moving due to the truck's acceleration.

Uniform Acceleration

Uniform acceleration occurs when an object's velocity changes at a constant rate. This means the speed increases (or decreases) steadily over time.
The key characteristics include:

• Constant acceleration value
• Linear velocity change

In the exercise, the truck accelerates uniformly from rest to 30.0 m/s.
Use the formula:
\[v = at\]
where

• \( v \): final velocity
• \( a \): acceleration
• \( t \): time

Here, you can calculate the shortest time the truck requires to reach a certain speed without the toolbox sliding, provided the static friction force is not exceeded. By understanding the principle of uniform acceleration, you can apply it to find how external forces, like static friction, affect the movement of objects on moving surfaces.