Question

Objects$\mathrm{A}$and$\mathrm{B}$are both positively charged. Both have a mass of$100\mathrm{g}$, but$\mathrm{A}$has twice the charge of B. When$\mathrm{A}$and $\mathrm{B}$are placed$10\mathrm{cm}$apart,$\mathrm{B}$ experiences an electric force of$0.45\mathrm{N}$.

a. What is the charge on$\mathrm{A}$?

b. If the objects are released, what is the initial acceleration of$\mathrm{A}$?

Short answer

a. Charge on$\mathrm{A}$is$1\times {10}^{-6}\mathrm{C}$.

b. The initial acceleration is$4.5\frac{\mathrm{m}}{{\mathrm{s}}^2}$.

Step-by-step solution

Calculation for charge (part a)

a.

The equation $\mathrm{F}=\frac{{\mathrm{kq}}_1{\mathrm{q}}_2}{{\mathrm{r}}^2}$used.

Where,

$\mathrm{k}=9\times {10}^9\frac{{\mathrm{Nm}}^2}{{\mathrm{C}}^2}$

Charge is,

${\mathrm{q}}_1\mathrm{F}=\frac{{\mathrm{kq}}_1\left(\frac{1}{2}{\mathrm{q}}_1\right)}{{\mathrm{r}}^2}$

$\to {\mathrm{Fr}}^2={\mathrm{kq}}_1\left(\frac{1}{2}{\mathrm{q}}_1\right)$

$\to 2{\mathrm{Fr}}^2={\mathrm{kq}}_1^2\to \sqrt{\frac{2{\mathrm{Fr}}^2}{\mathrm{k}}}$

$={\mathrm{q}}_1$

Where ${\mathrm{F}}_{\mathrm{AB}}={\mathrm{F}}_{\mathrm{BA}}=0.45\mathrm{N}$and $\mathrm{r}=10\mathrm{cm}=10\times {10}^{-2}\mathrm{m}$

So,

${\mathrm{q}}_1=\sqrt{\frac{2(0.45\mathrm{N}){\left(10\times {10}^{-2}\mathrm{m}\right)}^2}{9\times {10}^9\frac{{\mathrm{Nm}}^2}{{\mathrm{C}}^2}}}$

${\mathrm{q}}_1=1\times {10}^{-6}\mathrm{C}$

Calculation for initial acceleration (part b)

b.

Force is,

$\mathrm{F}=\mathrm{m}\times \mathrm{a}$

Acceleration is,

$\mathrm{a}=\frac{\mathrm{F}}{\mathrm{m}}$

So,

role="math" $\mathrm{a}=\frac{0.45\mathrm{N}}{0.1\mathrm{Kg}}$

$\mathrm{a}=4.5\frac{\mathrm{m}}{{\mathrm{s}}^2}$