Calculating Work Done By A Force: A Physics Problem

In the realm of physics, the concept of work is fundamental to understanding how energy is transferred and transformed. Work, in its simplest form, is the measure of energy transfer that occurs when a force causes an object to move over a certain distance. This article delves into the calculation of work done by a force, specifically focusing on a scenario where a particle moves between two points under the influence of a constant force. We will explore the underlying principles, mathematical formulations, and practical applications of this concept. Understanding work done by a force is crucial for comprehending various physical phenomena, from the motion of celestial bodies to the operation of machines. So, let's embark on this journey to unravel the intricacies of work and its significance in the world around us.

Problem Statement

Consider a particle that undergoes displacement from an initial position denoted by the vector $\vec{r_1} = 2\hat{i} + 3\hat{j}\f$ meters to a final position represented by the vector $\vec{r_2} = 3\hat{i} + 2\hat{j}\f$ meters. During this displacement, a constant force, given by the vector $\vec{F} = 5\hat{i} + 5\hat{j}\f$ Newtons, acts upon the particle. Our objective is to determine the amount of work done by this force on the particle as it moves from its initial to its final position. This problem exemplifies a classic scenario in physics where the concept of work is applied. By solving this problem, we will gain a deeper understanding of how to calculate work done by a constant force and how it relates to the displacement of an object.

Theoretical Background

Work Done by a Constant Force

The work (W) done by a constant force ($\vec{F}\f$) on an object as it undergoes a displacement ($\vec{d}\f$) is defined as the dot product of the force and displacement vectors. Mathematically, this is expressed as:

[ W = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos(\theta) ]

where:

• $|\vec{F}|\f$ is the magnitude of the force vector,
• $|\vec{d}|\f$ is the magnitude of the displacement vector, and
• $\theta\f$ is the angle between the force and displacement vectors.

The dot product effectively captures the component of the force that acts in the direction of the displacement. If the force and displacement are in the same direction ($\theta = 0^\circ\f$), the work done is maximum ($W = |\vec{F}| |\vec{d}|\f$). If they are perpendicular ($\theta = 90^\circ\f$), no work is done ($W = 0\f$). If the force and displacement are in opposite directions ($\theta = 180^\circ\f$), the work done is negative ($W = -|\vec{F}| |\vec{d}|\f$), indicating that the force is opposing the motion.

The unit of work in the International System of Units (SI) is the joule (J), which is defined as the work done by a force of one newton when it displaces an object by one meter in the direction of the force (1 J = 1 N⋅m).

Displacement Vector

The displacement vector ($\vec{d}\f$) represents the change in position of an object. It is the vector that points from the object's initial position to its final position. In terms of position vectors, the displacement vector is given by:

[ \vec{d} = \vec{r_2} - \vec{r_1} ]

where:

• $\vec{r_1}\f$ is the initial position vector, and
• $\vec{r_2}\f$ is the final position vector.

The displacement vector provides information about both the distance and direction of the object's movement. It is a crucial quantity in physics, particularly in the study of motion and work.

Solution

To find the work done by the force on the particle, we need to follow these steps:

1. Determine the displacement vector ($\vec{d}\f$):

   The displacement vector is the difference between the final and initial position vectors:

   [ \vec{d} = \vec{r_2} - \vec{r_1} = (3\hat{i} + 2\hat{j}) - (2\hat{i} + 3\hat{j}) ]

   [ \vec{d} = (3 - 2)\hat{i} + (2 - 3)\hat{j} = \hat{i} - \hat{j} \text{ m} ]

2. Calculate the work done ($W\f$):

   The work done is the dot product of the force vector ($\vec{F}\f$) and the displacement vector ($\vec{d}\f$):

   [ W = \vec{F} \cdot \vec{d} = (5\hat{i} + 5\hat{j}) \cdot (\hat{i} - \hat{j}) ]

   [ W = (5 \times 1) + (5 \times -1) = 5 - 5 = 0 \text{ J} ]

Therefore, the work done by the force on the particle is 0 Joules.

Detailed Explanation of the Solution

Step 1: Finding the Displacement Vector

In this crucial first step, we determine the displacement vector, which essentially represents the change in the particle's position. The displacement vector is calculated by subtracting the initial position vector ($\vec{r_1}\f$) from the final position vector ($\vec{r_2}\f$). This calculation gives us a vector that points from the particle's starting point to its ending point, providing both the distance and direction of the particle's movement. In our specific problem, we have:

[ \vec{r_1} = 2\hat{i} + 3\hat{j} \text{ m} ]

[ \vec{r_2} = 3\hat{i} + 2\hat{j} \text{ m} ]

Subtracting $\vec{r_1}\f$ from $\vec{r_2}\f$, we get:

[ \vec{d} = \vec{r_2} - \vec{r_1} = (3\hat{i} + 2\hat{j}) - (2\hat{i} + 3\hat{j}) ]

To perform the subtraction, we group the components along the $\hat{i}\f$ and $\hat{j}\f$ directions:

[ \vec{d} = (3 - 2)\hat{i} + (2 - 3)\hat{j} = 1\hat{i} + (-1)\hat{j} ]

Thus, the displacement vector is:

[ \vec{d} = \hat{i} - \hat{j} \text{ m} ]

This vector tells us that the particle moved 1 meter in the positive $x\f$-direction ($\hat{i}\f$) and 1 meter in the negative $y\f$-direction ($-\hat{j}\f$). The displacement vector is a fundamental quantity in physics as it directly relates to the work done by a force, as we will see in the next step.

Step 2: Calculating the Work Done

The work done by a constant force is calculated using the dot product of the force vector and the displacement vector. The dot product is a mathematical operation that takes two vectors and returns a scalar, which in this case represents the amount of energy transferred by the force during the displacement. The formula for the dot product is:

[ W = \vec{F} \cdot \vec{d} ]

where $W\f$ is the work done, $\vec{F}\f$ is the force vector, and $\vec{d}\f$ is the displacement vector. In our problem, we are given the force vector as:

[ \vec{F} = 5\hat{i} + 5\hat{j} \text{ N} ]

and we calculated the displacement vector in the previous step as:

[ \vec{d} = \hat{i} - \hat{j} \text{ m} ]

To compute the dot product, we multiply the corresponding components of the two vectors and then add the results. This is done as follows:

[ W = (5\hat{i} + 5\hat{j}) \cdot (\hat{i} - \hat{j}) = (5 \times 1) + (5 \times -1) ]

Performing the multiplication and addition:

[ W = 5 - 5 = 0 ]

Therefore, the work done by the force on the particle is:

[ W = 0 \text{ J} ]

This result indicates that, although there was a force acting on the particle and the particle underwent displacement, no work was done. This might seem counterintuitive at first, but it occurs because the force and displacement vectors are orthogonal, meaning they are perpendicular to each other. In such cases, the force does not contribute to the motion along the displacement direction, and hence, no work is done. This outcome underscores an important principle in physics: work is only done when there is a component of the force acting along the direction of the displacement. The dot product mathematically captures this principle, providing a precise way to calculate work done in various physical scenarios.

Implications of Zero Work Done

The result of zero work done in this scenario carries significant implications in physics. It highlights the fact that the mere presence of a force and displacement does not guarantee that work is being done. Work, in the physics context, is a specific measure of energy transfer that occurs when a force causes displacement in its direction. In our case, the force vector $\vec{F} = 5\hat{i} + 5\hat{j}\f$ N and the displacement vector $\vec{d} = \hat{i} - \hat{j}\f$ m are such that their dot product is zero. This means that the force is acting in a direction that is perpendicular to the displacement. Geometrically, this can be visualized as the force vector and the displacement vector forming a right angle. When the force and displacement are perpendicular, the force does not contribute to the motion of the particle along the direction of displacement, and thus, no energy is transferred.

Scenarios with Zero Work

There are several physical scenarios where zero work is done, which can be categorized as follows:

1. Perpendicular Force and Displacement: The most straightforward case is when the force and displacement are perpendicular. A classic example is the centripetal force acting on an object in uniform circular motion. The centripetal force is always directed towards the center of the circle, while the displacement is tangential to the circle. Since these two vectors are always at right angles, the centripetal force does no work on the object, and the object's speed remains constant.

2. No Displacement: If an object does not move, no work is done, regardless of the force applied. For instance, if you push against a wall, you exert a force, but since the wall does not move, the displacement is zero, and thus the work done is zero.

3. Forces Cancel Each Other: In some situations, multiple forces may act on an object, but their combined effect results in no net force in the direction of displacement. This can occur in static equilibrium, where all forces balance each other, leading to no net work done.

Importance of Understanding Zero Work

Understanding the concept of zero work is crucial for several reasons:

• Energy Conservation: It helps in the correct application of the work-energy theorem, which states that the net work done on an object is equal to the change in its kinetic energy. If the net work is zero, the kinetic energy remains constant, indicating that there is no change in the object's speed.
• Problem-Solving: Recognizing scenarios where work is zero simplifies problem-solving in mechanics. It allows physicists to focus on the forces and displacements that contribute to energy transfer and motion.
• Conceptual Clarity: It enhances a deeper understanding of the physical meaning of work, which is not merely the application of force but the transfer of energy due to a force causing displacement in its direction.

In the context of our problem, the zero work done suggests that the force $\vec{F}\f$ is not contributing to any change in the particle's kinetic energy. If the particle were moving under the influence of other forces as well, the total work done would be the sum of the work done by each force. However, the force $\vec{F}\f$ itself is neither adding nor subtracting energy from the particle's motion, emphasizing the directional nature of work and energy transfer.

Real-World Applications

The principles of work done by a force have far-reaching applications in various fields of science and engineering. Understanding how work is calculated and the factors that influence it is essential for designing efficient systems and predicting the behavior of physical systems. Here are some real-world applications where the concept of work is critical:

1. Mechanical Engineering

In mechanical engineering, the concept of work is fundamental in the design and analysis of machines and mechanical systems. Engineers use the principles of work to:

• Design Engines and Motors: The efficiency of engines and motors is directly related to the work they can perform. Engineers calculate the work done by the forces generated within the engine or motor to optimize their performance. This involves analyzing the forces exerted by pistons, turbines, and other components to ensure maximum energy conversion and output.
• Analyze Structures: When designing structures like bridges and buildings, engineers must consider the work done by various forces, including gravity, wind, and applied loads. Understanding how these forces interact and distribute work helps in creating stable and safe structures.
• Optimize Machines: The operation of machines often involves repetitive motions and energy transfers. By calculating the work done in each cycle, engineers can optimize the design to reduce energy losses and improve overall efficiency. This is crucial in industries where energy consumption is a significant concern.

2. Robotics

In robotics, the concept of work is crucial for designing robots that can perform tasks efficiently and effectively. The calculation of work helps in:

• Motion Planning: Robots need to move objects and manipulate their environment. Calculating the work done by the robot's actuators (motors) helps in planning the most energy-efficient movements. This involves determining the forces required to move objects and the distances over which these forces must be applied.
• Force Control: Robots often need to exert specific forces to perform tasks, such as assembling parts or gripping objects. Understanding the work done by these forces is essential for precise force control and preventing damage to the robot or the environment.
• Energy Efficiency: Battery life is a critical consideration for mobile robots. By minimizing the work done in performing tasks, engineers can extend the robot's operational time. This involves optimizing the robot's movements and the forces it exerts to complete its tasks.

3. Biomechanics

Biomechanics applies the principles of mechanics to biological systems, including the human body. The concept of work is used to analyze:

• Muscle Activity: Understanding the work done by muscles during movement helps in analyzing human motion and designing rehabilitation programs. By calculating the forces exerted by muscles and the distances over which these forces act, biomechanists can assess muscle performance and identify areas for improvement.
• Joint Mechanics: The forces and torques acting at joints determine the work done during movements. Analyzing joint mechanics helps in understanding joint stability, load distribution, and the risk of injury. This is crucial in sports medicine and physical therapy.
• Ergonomics: Ergonomics aims to design workplaces and equipment that minimize physical strain and injury. Calculating the work done by workers in various tasks helps in identifying potential hazards and designing safer work environments.

4. Sports Science

In sports science, the concept of work is used to analyze athletic performance and improve training techniques. Applications include:

• Performance Analysis: Calculating the work done by athletes during activities like running, jumping, and throwing helps in quantifying their performance. This analysis can reveal strengths and weaknesses, guiding training strategies.
• Equipment Design: The design of sports equipment, such as shoes and athletic gear, often involves considering the work done by athletes. Optimizing equipment to reduce energy expenditure and improve performance requires a thorough understanding of work principles.
• Injury Prevention: Analyzing the forces and work done during sports activities helps in identifying mechanisms of injury. This knowledge is crucial for developing training programs and protective equipment that minimize the risk of injuries.

5. Civil Engineering

Civil engineers use the principles of work in various aspects of infrastructure design and maintenance:

• Construction Equipment: Operating heavy machinery, such as cranes and excavators, involves significant work. Civil engineers calculate the work done by these machines to ensure efficient operation and safety on construction sites.
• Structural Stability: Analyzing the work done by forces acting on structures, such as bridges and dams, is crucial for ensuring their stability and longevity. This involves considering factors like gravity, water pressure, and seismic forces.
• Material Testing: Testing the mechanical properties of materials often involves measuring the work done to deform or fracture them. This information is used to select appropriate materials for construction projects and ensure structural integrity.

Practical Examples

• Lifting Weights: When a weightlifter lifts a barbell, they are doing work against gravity. The amount of work done depends on the weight of the barbell and the height it is lifted.
• Pushing a Car: Pushing a stalled car requires doing work to overcome friction and inertia. The work done depends on the force applied and the distance the car is pushed.
• Pedaling a Bicycle: Pedaling a bicycle involves doing work to propel the bike forward. The work done depends on the force exerted on the pedals and the distance the bike travels.

These real-world applications demonstrate the broad relevance of the concept of work in physics and engineering. Understanding how to calculate work and its implications is essential for solving practical problems and designing efficient systems.

Conclusion

In summary, we have explored the concept of work done by a force, focusing on a scenario where a particle moves under the influence of a constant force. We began by defining work as the dot product of the force and displacement vectors, which captures the component of the force acting in the direction of motion. We then applied this definition to a specific problem, calculating the work done by a force $\vec{F} = 5\hat{i} + 5\hat{j}\f$ N as it displaced a particle from $\vec{r_1} = 2\hat{i} + 3\hat{j}\f$ m to $\vec{r_2} = 3\hat{i} + 2\hat{j}\f$ m. The solution revealed that the work done was 0 J, illustrating a crucial principle: work is zero when the force and displacement are perpendicular.

Key Takeaways

• Work is a measure of energy transfer: It occurs when a force causes displacement in its direction.
• The dot product is essential for calculating work: $W = \vec{F} \cdot \vec{d}\f$ provides the mathematical framework for determining work done by a force.
• Zero work does not imply zero force or displacement: It indicates that the force and displacement are orthogonal, and there is no energy transfer due to the force along the direction of displacement.
• Real-world applications of work are vast: From mechanical engineering to sports science, the concept of work is fundamental in analyzing and optimizing systems.

Importance of Understanding Work

Understanding the concept of work is essential for a comprehensive grasp of physics and its applications. It forms the basis for numerous physical principles and is crucial in fields such as engineering, robotics, and biomechanics. By mastering the calculation and interpretation of work, students and professionals can:

• Solve complex problems in mechanics: The work-energy theorem, which relates work done to changes in kinetic energy, is a powerful tool for analyzing motion.
• Design efficient systems: Engineers use work calculations to optimize machines, engines, and structures for maximum performance and energy efficiency.
• Analyze biological movements: Biomechanists apply the principles of work to understand muscle activity, joint mechanics, and human motion.
• Enhance athletic performance: Sports scientists use work analysis to improve training techniques and equipment design.

Final Thoughts

In conclusion, the concept of work is a cornerstone of physics, with far-reaching implications in both theoretical understanding and practical applications. By thoroughly grasping the principles of work, we can better analyze and design the world around us, optimizing systems for efficiency and performance. The exploration of this concept not only enhances our knowledge of physics but also equips us with valuable tools for problem-solving and innovation in diverse fields.