Q. 205.0( 3 Votes )

Consider the situation of the previous problem.

(a) Find the tension in the string in equilibrium.

(b) Suppose the ball is slightly pushed aside and released. Find the time period of the small oscillations.

Answer :


Given:


Length of silk thread=10cm


Mass of ball=10g


Charge of ball=4.0× 10-6C


Equilibrium angle of thread with vertical=60°


Let the surface charge density on the plate be σ



The forces acting on the ball are


• Weight of the ball downwards (W=mg)


Where m=mass of ball


g=acceleration due to gravity


• Electric force due to non-conducting plate producing electric field E (F=qE)


Where q=charge of the ball


E=electric field intensity


• Tension force (T) due to string at an angle of 60° from vertical


Electric field due to a thin non-conducting plate of surface charge density σ is given by


……(i)


The tension force T due to string is divided into horizontal and vertical components given by Tsin60° and Tcos60°


Since the ball is in equilibrium, the net horizontal and vertical force on the ball is zero


Applying equilibrium along horizontal direction, we get


…(ii)


Similarly, applying equilibrium along vertical direction, we get


….(iii)


Dividing eqn. (ii) by (iii)




Putting the value of E from eqn.(i), we get,




Putting the values of ϵ0,m,g and q in the above equation



C/m2


(a)Now to find the tension in the string in equilibrium condition we can use eqn.(iii)





N


Therefore tension in the string in equilibrium is given by 0.196N


(b)when the ball is slightly pushed aside and released, the ball will undergo small oscillations due to the restoring forces. When the ball will come in mean position, tension, weight and electric force will balance.



The forces acting on the ball are mg in vertical direction and qE in horizontal direction


From the fig, tension is given by



So the net acceleration geff is given by




Therefore time period of small oscillations T is given by




Putting the value of E from eqn.(i) we get



Putting the values of g, l, q, σ ϵ0 and m and solving we get,



Therefore the time period of small oscillations of ball is given by 0.45sec


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