Problem 11 - College Physics

Question

\(\cdot\) \(\cdot\) A boxed 10.0 kg computer monitor is dragged by friction 5.50 m up along the moving surface of a conveyor belt inclined at an angle of \(36.9^{\circ}\) above the horizontal. If the monitor's speed is a constant \(2.10 \mathrm{cm} / \mathrm{s},\) how much work is done on the monitor by (a) friction, (b) gravity, and (c) the normal force of the conveyor belt?

Step-by-Step Solution

Answer

Friction does negative work equal to gravity's work; normal force does zero work.

Step 1: Understand Work and Its Formula

The work done on an object is given by the formula: \( W = F \cdot d \cdot \cos(\theta) \), where \(F\) is the force applied, \(d\) is the distance moved by the object in the direction of the force, and \(\theta\) is the angle between the force and the direction of movement.

Step 2: Calculate Work Done by Gravity

The gravitational force can be found using \(F_g = m \cdot g\), where \(m = 10.0\, \text{kg}\) and \(g = 9.81\, \text{m/s}^2\). The component of the gravitational force along the incline is \(F_{g\parallel} = F_g \cdot \sin(36.9^\circ)\). The work done by gravity is \(W_g = F_{g\parallel} \cdot d\). Substitute the values to compute \(W_g\).

Step 3: Compute Work Done by Friction

Assuming the monitor moves at a constant speed, the net work done by all forces is zero (due to no change in kinetic energy). As a result, the work done by friction must be equal to and opposite in sign to the work done by both gravity and the conveyor. Calculate the work done by friction, knowing that it should balance the positive work done by gravity along the incline.

Step 4: Determine Work Done by Normal Force

The normal force acts perpendicular to the direction of movement on the incline. Since work is only done by a force component in the direction of motion, the work done by the normal force is zero \(W_n = 0\).

Step 5: Summarize the Calculations

Recall that work done by the friction and by the normal force are both related to the scenario of constant speed and incline. Summarize the work done by each force using the equations and solutions from previous steps.

Key Concepts

Friction

When pulling an object like a computer monitor up an inclined conveyor belt, friction plays a crucial role. Friction is the resistive force that acts opposite to the direction of motion, preventing slipping. In this case, since the monitor moves at a constant speed, friction should perfectly balance the other forces acting on it. Here are a few important features of friction:

• Direction: Friction always opposes motion. If the motion is upwards along the incline, friction acts downwards.
• Source: It originates from the contact between the monitor and the surface.
• Energy Loss: Friction converts some of the mechanical energy into heat.

The work done by friction equals the work done by other forces (like gravity) but with the opposite sign. This ensures constant speed with no net gain in kinetic energy.

Gravitational Force

Gravitational force is a fundamental concept that attracts objects with mass towards each other. For a monitor on an incline, gravity acts downwards. The force is defined by the equation: \( F_g = m \cdot g \), where \( m \) is mass (here 10 kg) and \( g \) is the acceleration due to gravity (approximately 9.81 m/s²). The work done by gravity along an inclined plane requires calculating the component of this force parallel to the incline, which we find using the sine of the angle:

• Constant and Predictable: Gravity remains constant and acts downwards.
• Works Parallel to Incline: By considering the angle of the incline (36.9° in this exercise), we find the effective force performing work parallel to the slope.

This parallel component determines the contribution of gravity's work along the slope.

Normal Force

The normal force is the one exerted perpendicular to a surface when an object is in contact with it. In the scenario of the monitor on the conveyor belt, the normal force acts perpendicular to the plane of the conveyor.

• Perpendicular Action: Normal forces act at a right angle to the surface.
• No Work Done: Because work is only accomplished by the force component in the direction of motion, the normal force performs no work here. The work done is zero, since there is no displacement along this force’s direction.
• Supports the Object: It balances the object against gravitational force pushing it into the surface.

The normal force is crucial to maintaining the object's position on the conveyor while not contributing to the work done.

Inclined Plane

An inclined plane simplifies the understanding of various force components acting on an object. When pulling a monitor up such a plane:

• Slope Influence: The slope angle affects how much of gravitational force contributes to pulling the object down the incline.
• Force Breakdown: Forces like gravity can be divided into components parallel and perpendicular to the plane.
• Reduced Force Needs: Inclines reduce the force necessary to lift an object compared to a vertical lift, making it easier to manage with less energy.

This setup on a conveyor belt helps visualize how different forces might work together and balance in practical scenarios, such as transporting goods or machinery moving uphill.

Constant Speed

Achieving constant speed when moving up an inclined plane means that all forces contributing to the motion are balanced. Constant speed indicates a state of equilibrium, where no net force is acting to accelerate or decelerate the object.

• Zero Net Work: The sum of work done by all forces equals zero, as there is no net change in kinetic energy.
• Friction Balance: Friction perfectly counters the forces propelling the monitor upwards, such as the engine force of the conveyor belt, and part of gravity parallel to the plane.
• Practical Applications: Maintaining constant speed is essential in industrial applications for safe and efficient operation.

Understanding constant speed on an incline provides insight into how forces must be equally balanced to maintain a steady state of motion.