Problem 85 - University Physics with Modern Physics

Question

A 40.0 -kg packing case is initially at rest on the floor of a 1500 -kg pickup truck. The coefficient of static friction between the case and the truck floor is \(0.30,\) and the coefficient of kinetic friction is \(0.20 .\) Before each acceleration given below, the truck is traveling due north at constant speed. Find the magnitude and direction of the friction force acting on the case (a) when the truck accelerates at 2.20 \(\mathrm{m} / \mathrm{s}^{2}\) northward and (b) when it accelerates at 3.40 \(\mathrm{m} / \mathrm{s}^{2}\) southward.

Step-by-Step Solution

Answer

The friction forces are static (northward) for (a) and kinetic (northward) for (b).

Step 1: Understand the Problem

We have a packing case on a truck floor that experiences friction when the truck accelerates. We want to find the friction force when the truck accelerates both northward and southward.

Step 2: Calculate Maximum Static Friction Force

The maximum static friction force that can act on the packing case is given by: \( f_s = \mu_s \cdot m \cdot g \) where \( \mu_s = 0.30 \), \( m = 40.0 \) kg, and \( g = 9.8 \) m/s². Calculate \( f_s \).

Step 3: Calculate Required Force for Northward Acceleration

When the truck accelerates northward at 2.20 m/s², find the force needed to accelerate the case: \( F = m \cdot a = 40.0 \times 2.20 \). Compare this with the maximum static friction force calculated.

Step 4: Determine Friction Force for Northward Acceleration

Since the required force to move the case (from the previous step) is less than the maximum static friction force, the static friction force equals the force needed to accelerate the case, northward.

Step 5: Calculate Static vs Kinetic Friction Force

Check if the required force for southward acceleration (calculated in a similar way as the northward case, but now with \( a = 3.40 \) m/s²) exceeds the maximum static friction. If it does, calculate the kinetic friction force using \( f_k = \mu_k \cdot m \cdot g \) where \( \mu_k = 0.20 \).

Step 6: Determine Friction Force for Southward Acceleration

If the required force exceeds the static friction, the case moves. Thus, the friction force is kinetic, acting northward (opposing the south acceleration). Calculate the kinetic friction force.

Key Concepts

Static Friction

Static friction is a force that resists the initiation of sliding motion between two surfaces. It acts when two surfaces are in contact but have not yet started to move relative to each other. This type of friction ensures that objects remain in their position until a sufficient force is applied to overcome it.

To calculate the maximum static friction force (\( f_s \)), you use the formula:

• \( f_s = \mu_s \cdot m \cdot g \)

Where \( \mu_s \) is the coefficient of static friction, \( m \) is the mass of the object, and\( g \) is the acceleration due to gravity (approximately \( 9.8 \ m/s^2 \)).

In our exercise, for the packing case with a mass of 40.0 kg and a static friction coefficient of 0.30, the calculation would proceed as:

• \( f_s = 0.30 \times 40.0 \times 9.8 = 117.6 \ N \)

This means that this amount of force is needed to start moving the case across the truck floor.

Static friction is critically important because it prevents slipping or sliding until the applied force surpasses this threshold. Thus, when the truck accelerates northwards at 2.20 m/s², if the force required to move the case is less than 117.6 N, the static friction will hold the case in place.

Kinetic Friction

Once an object starts moving, it is subject to kinetic friction. This type of friction acts in the direction opposite to the movement of the object, attempting to slow it down.

The calculation for kinetic friction force (\( f_k \)) is given by:

• \( f_k = \mu_k \cdot m \cdot g \)

where \( \mu_k \) is the coefficient of kinetic friction. Kinetic friction is usually less than static friction, which is why it is easier to keep an object moving once it has started.

In the exercise, the kinetic friction coefficient is 0.20, so the kinetic friction force when the case is moving would be:

• \( f_k = 0.20 \times 40.0 \times 9.8 = 78.4 \ N \)

If the force applied on the case exceeds the maximum static friction, the case starts moving, and kinetic friction takes over. This explains why, when the truck accelerates southward at 3.40 m/s², surpassing the static threshold, the friction force shifts to kinetic friction, acting in the direction opposite to the applied force to counteract the motion.

Newton's Second Law

Newton's Second Law provides the framework for understanding how forces affect the motion of objects. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The formula for this law is:

• \( F = m \cdot a \)

where \( F \) represents the force applied, \( m \) is the mass of the object, and\( a \) is the acceleration.

In our scenario with the pickup truck, we apply this principle to determine the force needed to accelerate the packing case in the truck.

• For northward acceleration at 2.20 m/s²:\( F = 40.0 \times 2.20 = 88 \ N \)
• For southward acceleration at 3.40 m/s²:\( F = 40.0 \times 3.40 = 136 \ N \)

These forces must be compared with static and kinetic friction forces to see whether the case remains stationary or starts moving. For the northward acceleration, since 88 N is less than the static friction threshold (117.6 N), the case does not move. However, for the southward acceleration, 136 N exceeds the static friction and triggers the kinetic friction, meaning the case moves, and friction acts opposite the motion direction to decelerate it.

Understanding Newton's Second Law in this context clarifies how different forces interact with the mass and acceleration of objects, providing a comprehensive view of motion dynamics.