A particle A, having a charge of 5.0*10^-7C is fixed in a vertical wall. A second particle B of mass 100 g and having equal charge is suspended by a silk thread of length 30 cm from the wall. The point of suspension is 30 cm above the particle A---?

(continued) Find the angle of the thread with the vertical when it stays in equilibrium.

Forces acting on the particle are
(i) weight mg downward
(ii) tension T along the thread
(iii) electric force of repulsion F

Coulomb's Law,

wikipedia

3 Answers
Aug 9, 2017

Let O be point of suspension of charged particle B. Charged particle A is fixed and particle B is held in equilibrium under forces as shown in the fig

  1. Coulomb's Force of repulsion F along AB
  2. Tension T in the silk thread.
  3. Weight acting downwards mg
    ![http://vidyadrishti.com](useruploads.socratic.org)

Let the thread make angle theta with the vertical.
We see that Delta AOB is an isosceles triangle.
Since its vertex angle =theta, base angles are =(90^@-theta/2). (Sum of all three angles of a triangle =180^@.)

If we drop a perpendicular from O on AB, it divides the line AB in two equal parts. Using trigonometry

AB=2xx[0.3xxsin(theta/2)]
AB=0.6sin(theta/2)m

The magnitude of Coulomb's Force is found from the given expression

F = k_e (|q_A q_B|)/ (AB)^2
where k_e is Coulomb's constant= 8.99×10^9 N m^2 C^-2

Inserting various values we get

F = 8.99×10^9xx (5.0xx10^-7xx5.0xx10^-7)/ (0.6sin(theta/2))^2
=>F = (6.24xx10^-3)/ (sin^2(theta/2))

Using Lami's theorem which relates the magnitudes of three coplanar, concurrent and non-collinear forces, T,F and W keeping the charged particle B in static equilibrium and angles between these we get
F/(sin(180-theta))=(mg)/(sin(90+theta/2))=T/(sin(90+theta/2))

Using first equality and simplifying we get

F/(sintheta)=(mg)/(cos(theta/2))

Taking g=9.81ms^-2, inserting various values and rewriting sin theta in terms of half angle we get
((6.24xx10^-3)/ (sin^2(theta/2)))/(2sin(theta/2)cos(theta/2))=(0.1xx9.81)/(cos(theta/2))
=>(6.24xx10^-3)/(2sin^3(theta/2))=0.1xx9.81
=>sin^3(theta/2)=(6.24xx10^-3)/(2xx0.1xx9.81)
=>sin^3(theta/2)=0.00318
=>sin(theta/2)=0.14706
=>(theta/2)=sin^-1 0.14706
=>theta=16.9^@

Aug 9, 2017

theta~~16.9164^@~~0.29525 rad

Explanation:

enter image source here

It is clear from the figure that Delta OAB is an isosceles triangle, so

2beta+theta=180^@
=>beta=90^@-theta/2

Also,

alpha+beta=90^@
=>alpha=90^@-(90^@-theta/2) = theta/2 ........................(1)

From figure it is clear that in equilibrium position,

mg*sin(theta)=F_e*cos(alpha)

=>mg*sin(2alpha)=F_e*cos(alpha)

=>2mg*sinalpha*cancelcolor(red)(cosalpha)=F_e*cancelcolor(red)(cos(alpha))

=>2mg*sinalpha=k_e(Q_AQ_B)/r^2=k_e(Q^2)/r^2...................(2)

Now from DeltaOAB,

costheta=(l^2+l^2-r^2)/(2*l^2)=1-(r^2)/(2l^2)

=>r^2=2l^2(1-costheta)=2l^2(1-cos(2alpha))

Put the value of r^2 in equation (2),

=>2mg*sinalpha=k_e(Q^2)/(2l^2(1-cos(2alpha)))=k_e(Q^2)/(2l^2(2-2cos^2alpha))

=>2mg*sinalpha=(k_eQ^2)/(4l^2sin^2alpha)

=>sinalpha=root(3)((k_eQ^2)/(8mgl^2))=root(3)((8.9875517873681764*10^9*(5*10^-7)^2)/(8*0.1*9.80665*(0.3)^2))= sin(theta/2)

=>theta=2*sin^-1(root(3)((8.9875517873681764*10^9*(5*10^-7)^2)/(8*0.1*9.80665*(0.3)^2)))

=>theta~~16.9164^@~~0.29525 rad

Aug 9, 2017

Alternate solution steps if you don't want to use Lami's Theorem

Explanation:

Let O be point of suspension of charged particle B. Charged particle A is fixed and particle B is held in equilibrium under forces as shown in the fig

  1. Coulomb's Force of repulsion F along AB
  2. Tension T in the silk thread.
  3. Weight acting downwards mg
    ![http://vidyadrishti.com](useruploads.socratic.org)

As three forces are in equilibrium, sum of vertical components of T and F are equal and opposite to mg and Horizontal components of T and F are equal and opposite. From the figure we see that F makes an angle =(90-theta/2) with the vertical and T makes an angle =theta with the vertical.

We get
Tcostheta+Fcos(90-theta/2)=mg
=>Tcostheta+Fsin(theta/2)=mg ..... (1)

Tsintheta=Fsin(90-theta/2) .....(2)
From (2)
=>T=1/sin theta Fcos(theta/2)
Inserting this value of T in (1) we get

(1/sin theta Fcos(theta/2))costheta+Fsin(theta/2)=mg
=>F (cos(theta/2)costheta+sin thetasin(theta/2))=mgsin theta

Using the trigonometric identity we get

F cos(theta-theta/2)=mgsin theta
=>Fcos(theta/2)=mgsin theta

Proceed as in other solution