Questions: Work from a Constant Force In this problem, you will calculate the work done by a constant force. A force is considered constant if F(r) is independent of r. This is the most frequently encountered situation in elementary Newtonian mechanics. Consider a particle moving in a straight line from initial point B to final point A, acted upon by a constant force F (Figure 1). In the figure, the force is indicated by a series of identical vectors pointing to the left, parallel to the horizontal axis. The vectors are all identical to reflect the idea that the force is constant everywhere along the path. The magnitude of the force is F, and the displacement vector from point B to point A is L (of magnitude L, making an angle θ (radians) with the positive x axis). Find WBA, the work that the force F performs on the particle as it moves from point B to point A. Express the work in terms of L, F, and θ. Remember to use radians, not degrees, for any angles that appear in your answer.

Work from a Constant Force In this problem, you will calculate the work done by a constant force. A force is considered constant if F(r) is independent of r. This is the most frequently encountered situation in elementary Newtonian mechanics. Consider a particle moving in a straight line from initial point B to final point A, acted upon by a constant force F (Figure 1). In the figure, the force is indicated by a series of identical vectors pointing to the left, parallel to the horizontal axis. The vectors are all identical to reflect the idea that the force is constant everywhere along the path. The magnitude of the force is F, and the displacement vector from point B to point A is L (of magnitude L, making an angle θ (radians) with the positive x axis). Find WBA, the work that the force F performs on the particle as it moves from point B to point A. Express the work in terms of L, F, and θ. Remember to use radians, not degrees, for any angles that appear in your answer.
Transcript text: Work from a Constant Force In this problem, you will calculate the work done by a constant force. A force is considered constant if $\vec{F}(\vec{r})$ is independent of $\vec{r}$. This is the most frequently encountered situation in elementary Newtonian mechanics. Consider a particle moving in a straight line from initial point $B$ to final point $A$, acted upon by a constant force $\vec{F}$ (Figure 1). In the figure, the force is indicated by a series of identical vectors pointing to the left, parallel to the horizontal axis. The vectors are all identical to reflect the idea that the force is constant everywhere along the path. The magnitude of the force is $F$, and the displacement vector from point B to point A is $\vec{L}$ (of magnitude $L$, making an angle $\theta$ (radians) with the positive $x$ axis). Find $W_{\mathrm{BA}}$, the work that the force $\vec{F}$ performs on the particle as it moves from point $B$ to point $A$. Express the work in terms of $L, F$, and $\theta$. Remember to use radians, not degrees, for any angles that appear in your answer.
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Solution

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Solution Steps

Step 1: Understand the Problem

We need to calculate the work done by a constant force \(\vec{F}\) on a particle moving from point B to point A. The force \(\vec{F}\) is constant and acts parallel to the horizontal axis. The displacement vector from B to A is \(\vec{L}\) and makes an angle \(\theta\) with the positive x-axis.

Step 2: Recall the Work Formula

The work \(W\) done by a force \(\vec{F}\) over a displacement \(\vec{L}\) is given by the dot product: \[ W = \vec{F} \cdot \vec{L} \]

Step 3: Apply the Dot Product Formula

The dot product of two vectors \(\vec{F}\) and \(\vec{L}\) is: \[ \vec{F} \cdot \vec{L} = F L \cos(\theta) \] where \(F\) is the magnitude of the force, \(L\) is the magnitude of the displacement, and \(\theta\) is the angle between the force and the displacement.

Final Answer

\[ W_{B \to A} = F L \cos(\theta) \]

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