Problem 33 - College Physics

Question

The coefficient of kinetic friction between a \(40 \mathrm{~kg}\) crate and the warehouse floor is \(70 \%\) of the corresponding coefficient of static friction. The crate falls off a forklift that is moving at \(3 \mathrm{~m} / \mathrm{s}\) and then slides along the warehouse floor for a distance of \(2.5 \mathrm{~m}\) before coming to rest. What is the coefficient of static friction between the crate and the floor?

Step-by-Step Solution

Answer

The coefficient of static friction is approximately 0.2624.

Step 1: Understanding the Problem

We need to calculate the coefficient of static friction (\( \mu_s \)) based on the information given about the kinetic friction and the motion of the crate.

Step 2: Calculate Deceleration from Slide

Given the crate slides a distance of \( 2.5 \text{ m} \) after falling off with an initial velocity of \( 3 \text{ m/s} \) and comes to rest, we can use the formula \( v^2 = u^2 + 2as \) to find the deceleration \( a \). Here, \( v = 0 \), \( u = 3 \text{ m/s} \), and \( s = 2.5 \text{ m} \). Substituting these values gives:\[ 0 = (3)^2 + 2a(2.5) \]\[ 0 = 9 + 5a \]\[ a = -\frac{9}{5} = -1.8 \text{ m/s}^2 \]

Step 3: Relating Deceleration to Kinetic Friction

The force causing the deceleration is due to kinetic friction. We use the formula \( F = ma = \mu_k mg \), where \( \mu_k \) is the coefficient of kinetic friction and \( g = 9.8 \text{ m/s}^2 \). Substituting \( a = -1.8 \text{ m/s}^2 \):\[ 40(-1.8) = \mu_k (40)(9.8) \]\[ \mu_k = \frac{1.8}{9.8} \approx 0.1837 \]

Step 4: Determine Static Friction Coefficient

We are given that the kinetic friction coefficient is \( 70\% \) of the static friction coefficient, i.e., \( \mu_k = 0.7\mu_s \). We can now express this as:\[ \mu_s = \frac{\mu_k}{0.7} \]Substituting the kinetic friction value:\[ \mu_s = \frac{0.1837}{0.7} \approx 0.2624 \]

Step 5: Final Answer

The coefficient of static friction between the crate and the floor is approximately \( 0.2624 \).

Key Concepts

Kinetic Friction

Kinetic friction is a force that opposes the motion of two surfaces sliding past each other. In this lesson, imagine pushing a crate across a floor. As you push, kinetic friction acts in the opposite direction of your push. This force is crucial because it determines how quickly an object slows down once it's set into motion.

• The magnitude of kinetic friction depends on two factors: the nature of the surfaces in contact and the normal force, which is the force perpendicular to the contact surface.
• The coefficient of kinetic friction (\(\mu_k\)) is a dimensionless value that represents how much frictional force will be generated between different materials.
• This coefficient is generally less than the coefficient of static friction, meaning it's easier to keep an object moving than to start moving it from rest.

Understanding kinetic friction helps in predicting the motion of objects and is essential for solving many physics problems involving moving objects.

Deceleration Calculation

Calculating deceleration is all about understanding how quickly an object slows down. In our exercise, a crate falls off a forklift at an initial speed and then stops after sliding a certain distance.

• The formula used for calculating deceleration when an object comes to rest is given by \(v^2 = u^2 + 2as\).
• Here, \(v\) is the final velocity (0, because the crate stops), \(u\) is the initial velocity, \(a\) is the acceleration (or deceleration, which is negative in slowing down), and \(s\) is the distance over which the object slows down.

Using this formula helps to find out how forces like friction affect the movement of objects and enables the prediction of how different materials interact when moving.

Physics Problem Solving

Physics problem solving involves a structured approach to understanding the principles at play in given situations, like our crate falling off a forklift.

• First, clearly understand all given information and what the problem is asking for.
• Identify the underlying physics concepts, such as friction, motion, or force, involved in the scenario.
• Apply relevant equations and mathematical formulas, such as the equations of motion or frictional force equations, to solve for the unknown variables.
• Finally, reason through your solution to ensure it makes sense in the context of the problem and check your calculations for accuracy.

Developing these skills is key to solving a wide range of physics problems effectively and efficiently.

Force and Motion

Force and motion are fundamental concepts in physics that describe how objects move and interact. When discussing the crate and the warehouse floor, these concepts explain how and why the crate decelerates after falling.

• Force is an interaction that changes the motion of an object - it can start, stop, or change the direction or speed of an object.
• Newton's Second Law of Motion, \(F = ma\), tells us that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
• In our scenario, the force of friction is acting opposite to the motion, causing the crate to decelerate.
• Understanding the relationship between force and motion helps us predict how objects will move under various conditions.

These principles not only apply to crates in a warehouse but are also essential in many real-world applications, from designing vehicles to building infrastructure.