Question

A bolt is threaded onto one end of a thin horizontal rod, and the rod is then rotated horizontally about its other end. An engineer monitors the motion by flashing a strobe lamp onto the rod and bolt, adjusting the strobe rate until the bolt appears to be in the same eight places during each full rotation of the rod (Fig. 6-42). The strobe rate is$\mathbf{2000}$flashes per second; the bolt has mass$\mathbf{30}\mathit{g}$and is at radius$\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{cm}$. What is the magnitude of the force on the bolt from the rod?

Short answer

The magnitude of the force is $2.6\times {10}^3\text{\hspace{0.17em}N}$.

Step-by-step solution

Given data

• Mass of bolt,$\mathrm{m}=30.0\text{g}$ .
• Radius,$\mathrm{r}=3.5\text{cm}$
• Strobe rate,$2000\text{flashespersecond}.$

To understand the concept

The problem deals with the centripetal force. It is a force that makes a body follow a curved path.From the given information, the time period can be calculated. The force on the bolt from the rod is the centripetal force which can be calculated using its formula.

Formula:

Centripetal force is given by,

$\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$

Calculate the centripetal force 

We note that the period$\mathrm{T}$is eight times the time between flashes$\left(\frac{1}{2000}\text{\hspace{0.17em}s}\right)$.

So $\mathrm{T}=0.0040\text{s}$.

We know that,

$\mathrm{T}=\frac{2\mathrm{\pi r}}{\mathrm{v}}$

Also, centripetal force is given by,

$\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$

Combining the above two equations leads to,

$\mathrm{F}=\frac{4{\mathrm{m\pi }}^2\mathrm{R}}{{\mathrm{T}}^2}$

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

$\begin{array}{l}\mathrm{F}=\frac{4\left(0.030\text{\hspace{0.17em}kg}\right){\mathrm{\pi }}^2\left(0.035\text{\hspace{0.17em}m}\right)}{{\left(0.0040\text{\hspace{0.17em}s}\right)}^2}\\ \mathrm{F}=2.6\times {10}^3\text{\hspace{0.17em}N}\end{array}$

Thus, the magnitude of the force is $2.6\times {10}^3\text{\hspace{0.17em}N}$.