Problem 81 - University Physics with Modern Physics

Question

A steel washer is suspended inside an empty shipping crate from a light string attached to the top of the crate. The crate slides down a long ramp that is inclined at an angle of \(37^{\circ}\) above the horizontal. The crate has mass 180 \(\mathrm{kg}\) . You are sitting inside the crate (with a flashlight); your mass is 55 \(\mathrm{kg}\) . As the crate is sliding down the ramp, you find the washer is at rest with respect to the crate when the string makes an angle of \(68^{\circ}\) with the top of the crate. What is the coefficient of kinetic friction between the ramp and the crate?

Step-by-Step Solution

Answer

The coefficient of kinetic friction is approximately 0.466.

Step 1: Identify Key Components

We start by identifying the forces acting on the system. The forces involved include gravitational force on the crate, frictional force between the crate and the ramp, normal force, and tension in the string that is affected by the angle the washer makes inside the crate.

Step 2: Draw Free Body Diagram

Visualize the problem by sketching a free body diagram for the crate. Label the gravitational force acting down the ramp, the normal force perpendicular to the ramp, and the frictional force opposing the motion. The washer inside also helps to form another diagram as it rests at an angle from the vertical inside the accelerating frame.

Step 3: Analyze Forces on Crate

Calculate the gravitational force component parallel to the ramp:\[ F_{g, parallel} = (m_{crate} + m_{you}) \cdot g \cdot \sin(37^{\circ}) \]and perpendicular to the ramp:\[ F_{g, perpendicular} = (m_{crate} + m_{you}) \cdot g \cdot \cos(37^{\circ}) \]

Step 4: Analyze Forces on Washer

Because the washer is at rest relative to the crate, it experiences a force balance in the frame of reference moving with the crate. The equilibrium forces in the horizontal direction provide:\[ T \cdot \sin(68^{\circ}) = (m_{washer}) \cdot a \] and vertically:\[ T \cdot \cos(68^{\circ}) = (m_{washer}) \cdot g \] From this, express acceleration \(a\) in terms of known quantities.

Step 5: Calculate Acceleration

Use the washer's equilibrium relation to find the acceleration: From the vertical force balance, \[ m_{washer} \cdot a = g \cdot \tan(68^{\circ}) \]This gives us the acceleration of the crate \(a_x\) in terms of gravity and angle.

Step 6: Combine Positive and Friction Forces

The acceleration along the ramp combined with static equilibrium of forces gives:\[ F_{net} = (m_{crate} + m_{you}) \cdot a_x = F_{g, parallel} - F_{friction} \]Solving for friction, \( F_{friction} = \mu_k \cdot F_{g, perpendicular} \). Equating the net force and friction will allow us to solve for the coefficient of kinetic friction \(\mu_k\).

Step 7: Compute the Coefficient of Kinetic Friction

After substituting the value of \(a_x\) obtained earlier and simplifying with the force equations:\[ \mu_k = \frac{ F_{g, parallel} - (m_{crate} + m_{you}) \cdot a_x }{ F_{g, perpendicular} } \]Plug in the numbers to find \(\mu_k\).

Key Concepts

Free Body Diagram

A Free Body Diagram is a simple sketch used to visualize all the forces acting on an object. It's a crucial step in solving physics problems. In this specific exercise, we dealt with a crate sliding down an inclined plane and how it affects a suspended washer inside it. By drawing a free body diagram, we could identify each force and understand how they interact with one another.

Here are the key forces you would typically depict with arrows in a diagram:

• Gravitational force, pulling the crate down the ramp.
• Normal force, perpendicular to the surface of the inclined plane.
• Frictional force, opposing the slide down the ramp.
• Tension in the string, holding the washer at an angle inside the crate.

Visualizing these forces allows a step-by-step approach to solve the problem by setting up equations that describe the physical scenario. It helps in breaking down complex problems into manageable pieces.

Kinetic Friction

Kinetic friction is the force that opposes the relative motion of two surfaces sliding past each other. In this context, it acts between the crate and the inclined ramp. Understanding kinetic friction is key to resolving problems involving movement across surfaces.

Here’s what you need to know about kinetic friction:

• It depends on both the nature of the surfaces in contact and the normal force between them.
• The coefficient of kinetic friction, denoted by \(\mu_k\), is a dimensionless value that represents the friction's strength.

In the exercise, the frictional force can be calculated using the equation:
\[ F_{friction} = \mu_k \cdot F_{g, perpendicular} \]
Where \( F_{g, perpendicular} \) is the component of gravitational force perpendicular to the ramp. Solving the problem involved balancing the forces to find this coefficient \(\mu_k\), critical for understanding how friction impacts the system's motion.

Inclined Plane

An inclined plane is a flat surface tilted at an angle, with one of its ends raised higher than the other, creating a slope. It is a basic mechanical concept that eases the movement of objects by distributing the force over a longer distance.

When analyzing problems involving an inclined plane:

• Consider the angle of incline, which influences how much force is needed to move objects along the slope.
• Break down the gravitational force into components parallel and perpendicular to the plane’s surface.

This exercise had us analyze a crate sliding down a ramp inclined at \(37^{\circ}\). For such problems, determine the gravitational forces acting along these two components:
1. **Parallel to the incline**: Calculates how the crate will accelerate down the slope:
\[ F_{g, parallel} = (\text{mass}) \cdot g \cdot \sin(37^{\circ}) \]
2. **Perpendicular to the incline**: Influences the normal force (and consequently the friction):
\[ F_{g, perpendicular} = (\text{mass}) \cdot g \cdot \cos(37^{\circ}) \]
Understanding the physics of inclined planes helps resolve complex interactions of forces that occur when objects face slopes.