Question

A horizontal force equal to the object's weight is applied to an object resting on a table. What is the acceleration of the moving object when the coefficient of kinetic friction between the object and floor is 1 (assuming the object is moving in the direction of the applied force)? a) zero b) \(1 \mathrm{~m} / \mathrm{s}^{2}\) c) Not enough information is given to find the acceleration.

Short answer

Answer: The acceleration of the object is zero.

Step-by-step solution

Understand the problem and draw a free body diagram

In this problem, we are given the coefficient of kinetic friction between the object and the floor (μ = 1). We also know that a horizontal force equal to the object's weight (F = mg) is applied on the object. We need to find the acceleration (a) that the object will experience. First, we can draw a free body diagram to understand the forces acting on the object while it's moving on the surface. We need to consider the gravitational force (mg), normal force (N), applied force (F), and kinetic frictional force (f).

Apply Newton's Second Law of Motion

The next step is to apply Newton's second law of motion, F = ma, to determine the net force acting on the object in both the horizontal and vertical directions. In the vertical direction, there is no acceleration, so the sum of the vertical forces must equal zero. Therefore, we have: N = mg. In the horizontal direction, the net force is the difference between the applied force (F) and the frictional force (f). Thus, we have: F - f = ma.

Calculate the Kinetic Frictional Force

We can now calculate the kinetic frictional force using the given value of the coefficient of kinetic friction (μ): f = μN. Since μ = 1 and N = mg, we have: f = mg.

Find the Acceleration

We can now substitute the frictional force (f) into the horizontal net force equation: F - mg = ma. Since F = mg, we get: mg - mg = ma. This simplifies to: 0 = ma. Finally, we see that the acceleration (a) must be 0, because 0 = ma. Answer: a) zero

Key concepts

Newton's Second Law of Motion

Newton's Second Law of Motion is foundational to our understanding of how forces interact with mass to create acceleration. It's elegantly stated as F = ma, where F represents the net force acting on an object, m the object's mass, and a its acceleration. In simpler terms, this law tells us that an object’s acceleration is directly proportional to the net force acting upon it and inversely proportional to its mass.

When multiple forces are acting on an object, like in our textbook problem, it's crucial to find the net force to predict the object's motion. If the net force is zero, the acceleration is zero, which means the object either remains at rest or moves with a constant velocity. Understanding this relationship allows students to solve complex problems by breaking them down into manageable parts and analyzing the forces involved.

Free Body Diagram

A free body diagram is a powerful tool in physics that helps visualize the forces acting on an object. The diagram involves a simple sketch where the object is represented as a dot or a simple shape, and all of the forces acting on it are drawn as arrows extending from the center of the object. Each arrow is labeled with the force it represents, such as gravitational force, normal force, applied force, and frictional force.

Creating a free body diagram is a critical step in solving physics problems because it allows students to focus on the forces that matter for the particular situation. After drawing these diagrams, students can better apply principles like Newton's Second Law to find the acceleration of the object by examining the balance or imbalance of forces represented in the diagram.

Coefficient of Kinetic Friction

The coefficient of kinetic friction, often denoted by the Greek letter μ, is a dimensionless value that represents the ratio of the force of kinetic friction between two objects to the normal force pressing them together. In the context of our problem, the coefficient is given as 1, which simplifies the calculations considerably because it means the force of kinetic friction is equal to the normal force.

The value of μ depends on the materials involved and their condition. For example, ice on metal would have a low coefficient of kinetic friction, while rubber on concrete would have a higher one. Understanding how μ influences the force of friction is key to solving many real-world problems involving motion and helps explain how different materials interact when they slide against one another.

Acceleration Due to Forces

In physics, acceleration is understood as any change in velocity, and it occurs whenever there is an unbalanced force acting on an object. According to Newton's Second Law, the acceleration of an object is directly proportional to the net force and inversely proportional to the mass (a = F/m). This principle is fundamental when determining how an object will move.

In the scenario from our textbook, the acceleration is found to be zero, which can initially be a surprising result. But upon closer examination, we see that the force of kinetic friction perfectly balances the applied force, making the net force zero. As a result, the object does not accelerate, underlining the importance of considering all forces, including friction, to understand an object's motion accurately.