Work Done

Definition

Work done is a physical quantity that represents the amount of energy transferred by a force acting on an object as the object moves in the direction of the force. It is a fundamental concept in physics that quantifies the interaction between a force and the displacement of an object.

5 Must Know Facts For Your Next Test

1. The work done by a force is equal to the product of the force and the displacement of the object in the direction of the force.
2. Work can be positive, negative, or zero, depending on the relative directions of the force and the displacement.
3. The dot product of two vectors is used to calculate the work done by a force acting on an object.
4. The work done by a constant force is the product of the force, the displacement, and the cosine of the angle between the force and the displacement.
5. The work done by a variable force is calculated by integrating the dot product of the force and the differential displacement over the entire path of the object.

Review Questions

• Explain how the dot product is used to calculate the work done by a force acting on an object.
  • The dot product of two vectors, $\vec{F}$ and $\vec{d}$, is used to calculate the work done by a force $\vec{F}$ acting on an object that undergoes a displacement $\vec{d}$. The work done is given by the formula $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$, where $\theta$ is the angle between the force and the displacement. This formula shows that the work done is proportional to the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between them. The dot product captures the component of the force that is in the direction of the displacement, which is the only component that contributes to the work done.
• Describe how the work done by a variable force is calculated.
  • When the force acting on an object is variable, the work done cannot be calculated simply by multiplying the force, displacement, and the cosine of the angle between them. Instead, the work done must be calculated by integrating the dot product of the force and the differential displacement over the entire path of the object. This is expressed mathematically as $W = \int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r}$, where $\vec{r}_1$ and $\vec{r}_2$ are the initial and final positions of the object, respectively, and $d\vec{r}$ is the differential displacement vector. This integral captures the cumulative work done by the variable force as the object moves along its path.
• Analyze the relationship between work done, force, and displacement, and explain how this relationship is affected by the angle between the force and displacement.
  • The work done by a force is directly proportional to the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between the force and the displacement. This relationship is captured by the formula $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$. When the force and displacement are in the same direction ($\theta = 0^{\circ}$), the cosine of the angle is 1, and the work done is maximized. When the force and displacement are in opposite directions ($\theta = 180^{\circ}$), the cosine of the angle is -1, and the work done is negative, indicating that the force is doing negative work on the object. When the force and displacement are perpendicular ($\theta = 90^{\circ}$), the cosine of the angle is 0, and the work done is zero, meaning the force is not contributing to the object's displacement.

Related terms

Force:

A vector quantity that represents the interaction between two objects, causing a change in the motion or shape of one or both objects.

Displacement:

The change in the position of an object, described by a vector quantity that represents the straight-line distance between the initial and final positions.

Energy: The capacity to do work or cause change, which can take various forms, such as kinetic, potential, thermal, or electrical energy.