Q9DQ - University Physics with Modern Physics

Question

A force $\overrightarrow{\mathbf{F}}$ is in the x-direction and has a magnitude that depends on x. Sketch a possible graph of F versus x such that the force does zero work on an object that moves from ${\mathit{x}}_{\mathbf{1}}$ to ${\mathit{x}}_{\mathbf{2}}$, even though the force magnitude is not zero at all x in this range.

Step-by-Step Solution

Answer

The graph is as follows.

Step 1: Describe the Work

The expression for the work is given by,

$\mathit{W}\mathbf{=}\overrightarrow{\mathbf{F}}\mathbf{.}\overrightarrow{\mathbf{s}}$ 

Here, is the force and is the displacement.

Step 2: Sketch a possible graph of F versus x such that the force does zero work

Sketch the graph as follows

The force is acting in opposite direction at points $x_1$ and  $x_2$ but the magnitude is same, then the work done is zero.

 

The work done from point $x_1$ to $x_2$ is given by,

$$\begin{gathered} W=Area of x_{1}A{x}_{2}B{x}_1 \\ =Area x_1{x}_{2}A+Area{x}_1{x}_{2}B \\ =\frac{1}{2}{f}_x\left(x_2-x_1\right)+\frac{1}{2}\left(-f_x\right)\left(x_2-x_1\right) \\ =0 \end{gathered}$$ 

 

Therefore, no forces act on the bob of the conical pendulum when it is swinging.