Problem 64 - College Physics

Question

In moving a 35.0 -kg desk from one side of a classroom to the other, a professor finds that a horizontal force of \(275 \mathrm{~N}\) is necessary to set the desk in motion, and a force of \(195 \mathrm{~N}\) is necessary to keep it in motion at a constant speed. What are the coefficients of (a) static and (b) kinetic friction between the desk and the floor?

Step-by-Step Solution

Answer

Static friction coefficient: 0.802; Kinetic friction coefficient: 0.569.

Step 1: Understand the Problem Statement

We are given two pieces of information: the force required to set the desk in motion (\( F_s = 275 \text{ N} \)) and the force required to keep it moving at a constant speed (\( F_k = 195 \text{ N} \)). The mass of the desk is given as \( m = 35.0 \text{ kg} \). We need to find the coefficients of static and kinetic friction.

Step 2: Calculate the Normal Force

The normal force (\( N \)) is the force exerted by a surface to support the weight of an object resting on it. Because the desk is on a horizontal surface and there is no vertical movement, the normal force equals the gravitational force on the desk, calculated as:\[ N = m \, g \]Where \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity.Plugging in the values: \[ N = 35.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 343 \text{ N} \]

Step 3: Calculate the Coefficient of Static Friction

The coefficient of static friction (\( \mu_s \)) is calculated using the formula:\[ F_s = \mu_s \, N \]Rearranging for \( \mu_s \): \[\mu_s = \frac{F_s}{N} = \frac{275 \, \text{N}}{343 \, \text{N}} \approx 0.802 \]

Step 4: Calculate the Coefficient of Kinetic Friction

The coefficient of kinetic friction (\( \mu_k \)) is calculated using the formula:\[ F_k = \mu_k \, N \]Rearranging for \( \mu_k \):\[\mu_k = \frac{F_k}{N} = \frac{195 \, \text{N}}{343 \, \text{N}} \approx 0.569 \]

Step 5: Conclusion

The coefficients of static and kinetic friction are thus approximately 0.802 and 0.569, respectively.

Key Concepts

Static Friction

Static friction is the force that resists the initial movement of an object that is at rest. It acts parallel to the surfaces in contact. You can think of static friction as the force you must overcome to get something moving. In our problem, the force needed to start moving the desk was 275 N. This force identifies the maximum force of static friction.

• This force happens because surfaces are never perfectly smooth, leading to tiny rough patches that "lock" together.
• Static friction prevents objects from moving until a threshold force is applied.

To find the coefficient of static friction (\(\mu_s\)), we use the formula: \[ F_s = \mu_s \cdot N \]Here, \(F_s\) is the force of static friction, and \(N\) is the normal force. In our task, calculating \(\mu_s\) gives us approximately 0.802. This coefficient is a ratio, indicating that the resisting frictional force is 80.2% of the normal force.

Kinetic Friction

While static friction involves getting an object moving, kinetic friction happens once the object is already in motion. In this situation, the kinetic friction is slightly less than static friction. It resists movement rather than stopping it from starting. In our example, a 195 N force was needed to keep the desk moving at a constant speed.

• Kinetic friction is often lower than static friction because there is less "locking" between surfaces once they are sliding past one another.
• This friction always acts opposite to the direction of motion.

To find the coefficient of kinetic friction (\(\mu_k\)), you apply:\[ F_k = \mu_k \cdot N \]By solving this equation, we obtained \(\mu_k \approx 0.569\). This means that kinetic friction is about 56.9% of the normal force.Understanding kinetic friction helps predict how much force is needed to maintain movement of an object.

Normal Force

The normal force (\(N\)) is crucial to solving friction problems. It is the force exerted by a surface to support the weight of an object resting on it. It's always perpendicular to the contact surface. In physics, it balances the gravitational force acting on an object on a horizontal surface, which explains why objects don’t fall through.

• The weight of the desk creates a downward force due to gravity.
• The normal force counteracts this force to prevent the desk from accelerating vertically.

In our problem, using the formula \(N = m \cdot g\) where \(m = 35.0 \,\text{kg}\) (mass of the desk) and \(g = 9.8 \, \text{m/s}^2\) (gravitational acceleration), we calculated \(N = 343 \, \text{N}\). This value is essential for determining both static and kinetic forces.

Physics Problem Solving

Solving physics problems is like a puzzle. It requires understanding concepts and applying them appropriately. For friction problems, understanding the types of friction and calculating the normal force is key.

• Begin by identifying forces involved, like in our task where static and kinetic friction were primary forces of focus.
• Recognize that balancing the equations of motion is fundamental. It helps to determine unknowns using known quantities.

In our specific example, the problem required calculating coefficients of friction. This involved translating word problems into mathematical equations, reflecting the physics laws in numbers. You must adapt your approach according to the context, knowing when and how to use particular formulas.

• Normal calculations occur frequently and help isolate other forces acting on a body.
• You'll use logic and algebra to rearrange formulas and find desired results.

Ultimately, problem-solving builds a deeper insight into how physical principles interplay with real-world scenarios.