Question

a charged sphere of mass 2.8 kg is suspended from a string. an electric force acts horizontally on the sphere so that the string makes an angle of 33 degree with the vertical when at rest. Find (a) the magnitude of the electric force and (b)the tension in the string.

Answer 1

The free-body diagram is drawn on the figure below:

Here $\overrightarrow{F_e}$ is the electric force, $\overrightarrow{T}$ is the tension force and $\overrightarrow{mg}$ is the gravity force.

Let's write down the Newton's second law in a projection on the x-axis:

$$F_e - T\sin\theta = 0$$

and on the y-axis:

$$T\cos \theta - mg = 0$$

From the last equation we can find the tension force:

$$T = \dfrac{mg}{\cos\theta} = \dfrac{2.8\cdot 9.81}{\cos33\degree} \approx 32.75N$$

Substituting the expression for $T$ into the first equation and expressing $F_e$, obtain:

$$F_e =T\sin\theta = \dfrac{mg}{\cos\theta}\sin \theta = mg\tan\theta$$

Thus:

$$F_e = 2.8\cdot 9.81\cdot \tan33\degree \approx 17.84 N$$

Answer. (a) the magnitude of the electric force is 17.84 N, (b) the tension in the string is 32.75 N.