A coin slides over a frictionless plane and across an xy coordinate system from the origin to a point with xy coordinates $\left(3.0m,4.0m\right)$ while a constant force acts on it. The force has magnitude 2.0 Nand is directed at a counterclockwise angle of${\mathbf{100}}^{\mathbf{°}}$ from the positive direction of the x axis. How much work is done by the force on the coin during the displacement?

The initial position of the coin$=\left(0,0\right)$.

The final position of the coin $=\left(3.0\mathrm{m},4.0\mathrm{m}\right)$.

Force is, $\mathrm{F}=20\mathrm{N}$.

The angle at which the force is directed is, $\mathrm{\theta }={100}^{°}$.

Using the formula for work in terms of force and displacement, the work done in the x and y directions can be found separately. After this, we can add them to get the total work done by the force on the coin.

Formula:

$$\begin{gathered} \mathrm{W}=\mathrm{F}∆\mathrm{x} \\ {\mathrm{W}}_{\mathrm{total}}={\mathrm{W}}_{\mathrm{x}}+{\mathrm{W}}_{\mathrm{y}} \end{gathered}$$

A free body diagram can be drawn as,

From the above diagram, we can calculate the work done in the x direction as,

${\mathrm{W}}_{\mathrm{x}}={\mathrm{F}}_{\mathrm{x}}∆\mathrm{x}$

Here,$∆\mathrm{x}$is the displacement in the x direction.

Substitute the values in the above expression, and we get,

$$\begin{gathered} {\mathrm{W}}_{\mathrm{x}}=\left(2.0\mathrm{N}\times \cos \left({100}^{°}\right)\right)\times \left(3.0\mathrm{m}-0\mathrm{m}\right) \\ {\mathrm{W}}_{\mathrm{x}}=-1.01\mathrm{N}.\mathrm{m} \\ {\mathrm{W}}_{\mathrm{x}}=-1.01\mathrm{J} \end{gathered}$$

From the above diagram, we can calculate the work done in the y direction as,

${\mathrm{W}}_{\mathrm{y}}={\mathrm{F}}_{\mathrm{y}}∆\mathrm{y}$

Here,$∆\mathrm{y}$is the displacement in the y direction.

Substitute the values in the above expression, and we get,

$$\begin{gathered} {\mathrm{W}}_{\mathrm{y}}=\left(2\mathrm{N}\times \sin \left({100}^{°}\right)\right)\times \left(4.0\mathrm{m}-0\mathrm{m}\right) \\ {\mathrm{W}}_{\mathrm{y}}=7.88\mathrm{N}.\mathrm{m} \\ {\mathrm{W}}_{\mathrm{y}}=7.88\mathrm{J} \end{gathered}$$

Total work done will be,

$\mathrm{W}={\mathrm{W}}_{\mathrm{x}}+{\mathrm{W}}_{\mathrm{y}}$

Substitute the values in the above expression, and we get,

$$\begin{gathered} \mathrm{W}=-1.01\mathrm{J}+7.88\mathrm{J} \\ \mathrm{W}=6.8\mathrm{J} \end{gathered}$$

6.8 J of work is done by the force of 2.0 N on a coin during displacement.