Problem 1 - Physics for Scientists and Engineers with Modern Physics

Question

The Problems in this Section are ranked \(1,\) II, or III according to estimated difficulty, with \((1)\) Problems being easiest. Level (III) Problems are meant mainly as a challenge for the best students, for "extra credit." The Problems are arranged by Sections, meaning that the reader should have read up to and inciuding that Section, but this Chapter also has a group of General Problems that are not arranged by Section and not ranked. \(\begin{array}{l}{\text { (1) If the coefficient of kinetic friction between a } 22 \text { -kg crate }} \\ {\text { and the floor is } 0.30 \text { , what horizontal force is required to }} \\ {\text { move the crate at a steady speed across the floor? What }} \\ {\text { horizontal force is required if } \mu_{k} \text { is zero? }}\end{array}\)

Step-by-Step Solution

Answer

The horizontal force required is 64.68 N; if \( \mu_k = 0 \), no force is needed.

Step 1: Identify Given Values and Concepts

We are given the mass of the crate, \( m = 22 \text{ kg} \), and the coefficient of kinetic friction, \( \mu_k = 0.30 \). We need to find the horizontal force required to move the crate at a steady speed, which means overcoming the force of kinetic friction. If \( \mu_k = 0 \), it means there is no friction.

Step 2: Calculate the Force of Kinetic Friction

The force of kinetic friction, \( f_k \), is calculated using the formula \( f_k = \mu_k \times N \), where \( N \) is the normal force. Since the crate is on a horizontal surface, \( N = mg \), where \( g \) is the acceleration due to gravity \( 9.8 \text{ m/s}^2 \). Thus, \( f_k = 0.30 \times 22 \times 9.8 \).

Step 3: Compute the Normal Force

First, calculate the normal force (\( N \)), which is equal to the weight of the crate: \( N = m \times g = 22 \times 9.8 = 215.6 \text{ N} \).

Step 4: Calculate Force Required for Steady Speed

Substitute the value of \( N \) back into the kinetic friction formula: \( f_k = 0.30 \times 215.6 = 64.68 \text{ N} \). To move at a steady speed, the applied horizontal force must equal \( 64.68 \text{ N} \).

Step 5: Evaluate Force with Zero Friction

If \( \mu_k = 0 \), then friction, \( f_k = 0 \). Therefore, the horizontal force required to maintain steady speed is also zero, assuming no other resistive forces.

Key Concepts

Normal Force

The normal force is a crucial factor when dealing with friction problems, particularly in physics. It represents the perpendicular force exerted by a surface on an object resting on it. In simpler terms, it's the support force that keeps the object from breaking through the surface or sinking into it. For example, imagine a crate resting on the ground – the normal force is what balances out the gravitational pull trying to push the crate downwards.

In mathematical terms, the normal force (\(N\)) is often equivalent to the weight of the object if the surface is horizontal and there are no vertical components of other forces acting on the object. You can calculate it using:

• \(N = m \times g\)

where \(m\) is the mass of the object and \(g\) represents the acceleration due to gravity, which is \(9.8\, \text{m/s}^2\).

Having a clear understanding of the normal force is essential for solving problems involving friction, as it directly affects the frictional force applied to the object.

Applied Force

The applied force in a physics context is a force that is applied to an object by a person or another object. When solving problems involving motion along a surface, it’s essential to understand how the applied force interacts with frictional forces.

• If the applied force is greater than the frictional force, the object will start moving, experiencing acceleration.
• If the applied force equals the frictional force, the object will move at a constant speed, as seen in the solution above.
• If less, the object will either stay still or decelerate if initially in motion.

In our specific problem, when the applied horizontal force equals the force of kinetic friction (calculated as \(64.68\, \text{N}\) when \(\mu_k = 0.30\)), the crate moves at a steady speed.

Understanding the balance between these forces is key to analyzing motion and predicting how changes in applied force affect an object's movement.

Coefficient of Friction

The coefficient of friction, denoted as \(\mu\), is a dimensionless value that represents the degree of interaction between two surfaces. There are two types: static (\(\mu_s\)) and kinetic (\(\mu_k\)). In the context of this problem, the focus is on kinetic friction, which comes into play when surfaces slide against each other.

The coefficient of friction is a variable that depends heavily on the textures and materials of the surfaces in contact. A higher coefficient indicates greater frictional resistance. For instance, rubber on concrete may have a high \(\mu\), while ice on metal has a much lower one.

To calculate the force of friction (\(f_k\)), the equation \(f_k = \mu_k \times N\) is used. This calculation incorporates both the coefficient of friction and the normal force previously discussed. This relationship emphasizes how the nature of surfaces and their alignment relative to gravity can influence the required force to move an object steadily.