Question

While a roofer is working on a roof that slants at 36\(^\circ\) above the horizontal, he accidentally nudges his 85.0-N toolbox, causing it to start sliding downward from rest. If it starts 4.25 m from the lower edge of the roof, how fast will the toolbox be moving just as it reaches the edge of the roof if the kinetic friction force on it is 22.0 N?

Short answer

The toolbox will be moving at approximately 5.23 m/s when it reaches the edge of the roof.

Step-by-step solution

Understand the Problem

The roofer's toolbox starts from rest on a roof with a 36-degree incline and slides downward. We need to find its speed at the edge of the roof, taking into account the forces acting on it, such as gravitational force and kinetic friction.

Identify Forces and Energy Concepts

The toolbox experiences gravitational force along the incline, which can be decomposed into components parallel and perpendicular to the incline. The kinetic friction opposes motion along the incline. We will use the work-energy principle to find the final velocity.

Calculate the Gravitational Force Component

The force due to gravity parallel to the incline is given by:\[ F_{\text{gravity}} = mg \cdot \sin(\theta) \]where \( m \) is the mass of the toolbox, \( g \) is the acceleration due to gravity (9.8 m/s²), and \( \theta \) is the angle of the incline. Thus: \[ F_{\text{gravity}} = 85.0 \cdot \sin(36^\circ) \approx 49.9 \text{ N} \]

Compute Net Force Along the Incline

The net force \( F_{\text{net}} \) acting on the toolbox along the incline is the gravitational force component minus the frictional force.\[ F_{\text{net}} = F_{\text{gravity}} - F_{\text{friction}} \]\[ F_{\text{net}} = 49.9 \text{ N} - 22.0 \text{ N} = 27.9 \text{ N} \]

Determine Toolbox Mass

The mass \( m \) of the toolbox is found by dividing its weight by the acceleration due to gravity:\[ m = \frac{85.0}{9.8} \approx 8.67 \text{ kg} \]

Use Work-Energy Principle

According to the work-energy principle, the work done by the net force is equal to the change in kinetic energy:\[ W = \Delta KE = \frac{1}{2} m v^2 - \frac{1}{2} m v_0^2 \]where \( v_0 = 0 \) (initial speed). The work done by the net force is:\[ W = F_{\text{net}} \times d = 27.9 \times 4.25 \approx 118.575 \text{ J} \]Set this equal to the kinetic energy to solve for final velocity \( v \):\[ 118.575 = \frac{1}{2} \cdot 8.67 \cdot v^2 \]

Solve for Final Velocity

Rearrange the equation to solve for \( v \):\[ v^2 = \frac{118.575 \times 2}{8.67} \]\[ v^2 \approx 27.348 \]\[ v \approx \sqrt{27.348} \approx 5.23 \text{ m/s} \]

Key concepts

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It acts in the opposite direction to the movement. In this scenario, the toolbox on the roof encounters kinetic friction as it slides down the inclined plane. The kinetic frictional force is given as 22.0 N. This force is crucial because it reduces the net force acting on the toolbox, thus affecting its acceleration down the incline.

Several factors determine the magnitude of kinetic friction:

• Nature of Surfaces: Different materials have different coefficients of friction, which influence the force.
• Normal Force: The force perpendicular to the surfaces in contact, which in this case is affected by the angle of the roof.

Kinetic friction is expressed mathematically as: \[ F_{\text{friction}} = \mu_k F_{\text{normal}}, \]where \( \mu_k \) is the coefficient of kinetic friction and \( F_{\text{normal}} \) is the normal force. Understanding this concept helps us predict how objects accelerate and the energy required to move them.

Gravitational Force

Gravitational force is a fundamental force that pulls objects toward the center of the Earth. For objects on an incline, gravity can be decomposed into two components: one that acts parallel to the slope and another that acts perpendicular to it.

In this problem, the component of gravitational force making the toolbox slide down the roof is important. This component is calculated using:\[ F_{\text{gravity\_parallel}} = mg \cdot \sin(\theta) \]where:

• \( m \) is the mass of the toolbox.
• \( g \) is the acceleration due to gravity, approximately 9.8 m/s².
• \( \theta \) is the angle of the incline (36 degrees).

In our case, this yields a force of approximately 49.9 N. This force is what initially sets the toolbox in motion, providing the energy required to overcome the frictional force. Only the component of gravitational force parallel to the incline influences the sliding motion.

Inclined Plane

An inclined plane is a flat surface tilted at an angle to the horizontal. It changes the effective force required to lift an object. In our scenario, the roof acts as an inclined plane with an angle of 36 degrees. This angle affects how forces such as gravity and friction interact with the toolbox.

Key characteristics of inclined planes:

• Angle of Incidence: This determines how steep the incline is, affecting both the gravitational force components and how easily an object slides down.
• Reduction of Required Effort: Inclined planes make it easier to move heavy objects vertically by distributing the weight over a longer distance.

Inclined planes help in applying the work-energy principle, as they allow for the calculation of potential energy changes and their conversion into kinetic energy. Such planes are common in real-life situations where tasks involve lifting or sliding loads, making them a hugely practical application of physics.