Chapter 5, Problem 1P

A certain orthodontist uses a wire brace to align a patient’s crooked tooth as in Figure P5.1. The tension in the wire is adjusted to have a magnitude of 18.0 N. Find the magnitude of the net force exerted by the wire on the crooked tooth.

Figure P5.1

To determine

The magnitude of the net force exerted by the wire on the crooked tooth.

Answer to Problem 1P

The magnitude of the net force exerted by the wire on the crooked tooth is $8.71\text{N}$.

Explanation of Solution

Given info: The tension in the wire is $18.0\text{N}$.

From the Figure, the vertical components of tension are acted inward direction in the wire brace and a normal force just acts upward to align the patient’s crooked tooth. Hence, this normal force represents the net force exerted by the wire on the crooked tooth.

The summation of the forces along $\text{y}$-axis is equal to the zero.

$\begin{array}{c}{\displaystyle \sum F_{\text{y}}}=0\\ N-T\sin 14°-T\sin 14°=0\end{array}$

Here,

$T$ is the tension the wire brace.

${\displaystyle \sum F_{\text{y}}}$ is summation of the forces along $\text{y}$-axis.

$N$ is the magnitude of the net force exerted by the wire on the crooked tooth.

Substitute $18.0\text{N}$ for $T$ in above equation to find $N$.

$\begin{array}{c}N-\left(18.0\text{N}\right)\sin 14°-\left(18.0\text{N}\right)\sin 14°=0\\ N=8.709\text{N}\\ \approx 8.71\text{N}\end{array}$

Conclusion:

Therefore, the magnitude of the net force exerted by the wire on the crooked tooth is $8.71\text{N}$.